# questions

tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-planoconvex-lens-has-thickness-of-4-cm-when-placed-on-a-horizontal-table-with-the-curved-surface-in-contact-with-it-the-apparent-depth-of-the-bottom

slug:
a-planoconvex-lens-has-thickness-of-4-cm-when-placed-on-a-horizontal-table-with-the-curved-surface-in-contact-with-it-the-apparent-depth-of-the-bottom-35013

question:
<div class="question">A planoconvex lens has thickness of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>cm</mi><mo>.</mo></math> When placed on a horizontal table, with the curved surface in contact with it, the apparent depth of the bottom most point of the lens is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>cm</mi><mo>.</mo></math> If the lens is inverted such that the plane face is in contact with the table, the apparent depth of the centre of the plane face is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>25</mn><mn>8</mn></mfrac><mo> </mo><mi>cm</mi><mo>.</mo></math> The focal length of the lens is (assume thickness of lens to be negligible)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>85</mn><mo> </mo><mi>cm</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>59</mn><mo> </mo><mi>cm</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>75</mn><mo> </mo><mi>cm</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cm</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>75</mn><mo> </mo><mi>cm</mi></math></span> </div>

answer:
<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/727ec91f-0e7a-4c2a-8986-84e1edf26463-image.png" style="width: 400px; height: 107px;"/></p><p>In first case refraction of the rays, takes place from a plane surface, so we can use <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mi>app</mi></msub><mo>=</mo><mfrac><msub><mi>d</mi><mtext>actual </mtext></msub><mi>μ</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>=</mo><mfrac><mn>4</mn><mi>μ</mi></mfrac><mo>⇒</mo><mi>μ</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mtext>app </mtext></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mtext>actual </mtext></msub><mo>=</mo></math> apparent and real depth, respectively.</p><p>Now, in second case, as from Fig. (b), refraction takes place from a spherical surface, so</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>μ</mi><mn>2</mn></msub><mi>v</mi></mfrac><mo>-</mo><mfrac><msub><mi>μ</mi><mn>1</mn></msub><mi>u</mi></mfrac><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>2</mn></msub><mo>-</mo><msub><mi>μ</mi><mn>1</mn></msub></mrow><mrow><mi>R</mi><mi>m</mi></mrow></mfrac></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mo>(</mo><mo>-</mo><mn>25</mn><mo>/</mo><mn>8</mn><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow><mrow><mo>(</mo><mo>-</mo><mn>4</mn><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mn>4</mn><mo>/</mo><mn>3</mn></mrow><mrow><mo>-</mo><mi>R</mi></mrow></mfrac></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>3</mn><mi>R</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mn>8</mn><mn>25</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>75</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>25</mn><mo> </mo><mi>cm</mi></math></p><p>Now, using Lens' Makers formula, to calculate focal length</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>=</mo><mfenced><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mfrac><mn>1</mn><msub><mi>R</mi><mn>1</mn></msub></mfrac><mo>-</mo><mfrac><mn>1</mn><msub><mi>R</mi><mn>2</mn></msub></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>=</mo><mfenced><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mfrac><mn>1</mn><mo>∞</mo></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mo>(</mo><mo>-</mo><mn>25</mn><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>75</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>75</mn><mo> </mo><mi>cm</mi></math></p></div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch1_P1.db
title:
a-planoconvex-lens-of-focal-length-30-cm-has-its-plane-surface-silvered-an-object-is-placed-40-cm-from-the-lens-on-the-convex-side-the-distance-of-the

slug:
a-planoconvex-lens-of-focal-length-30-cm-has-its-plane-surface-silvered-an-object-is-placed-40-cm-from-the-lens-on-the-convex-side-the-distance-of-the-74563

question:
<div class="question">A plano-convex lens of focal length $30 \mathrm{~cm}$ has its plane surface silvered An object is placed $40 \mathrm{~cm}$ from the lens on the convex side. The distance of the image from the lens is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$18 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$24 \mathrm{~cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$40 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$24 \mathrm{~cm}$</span> </div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
title:
a-point-particle-is-held-on-the-axis-of-a-ring-of-mass-m-and-radius-r-at-a-distance-r-from-its-centre-c-when-released-it-reaches-c-under-the-gravitati-1

slug:
a-point-particle-is-held-on-the-axis-of-a-ring-of-mass-m-and-radius-r-at-a-distance-r-from-its-centre-c-when-released-it-reaches-c-under-the-gravitati-1-19997

question:
<div class="question">A point particle is held on the axis of a ring of mass $m$ and radius $r$ at a distance $r$ from its centre C. When released, it reaches $\mathrm{C}$ under the gravitational attraction of the ring. Its speed at $C$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{\frac{2 \mathrm{G} m}{r}(\sqrt{2}-1)}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{\frac{\mathrm{G} m}{r}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{\frac{2 \mathrm{G} m}{r}\left(1-\frac{1}{\sqrt{2}}\right)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{\frac{2 \mathrm{G} m}{r}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{\frac{2 \mathrm{G} m}{r}\left(1-\frac{1}{\sqrt{2}}\right)}$</span> </div>

answer:
<div class="solution">Let ' $\mathrm{M}$ ' be the mass of the particle
<br/> <br/>Now, $E_{\text {initial }}=E_{\text {final }}$
<br/> <br/>i.e. $\frac{\mathrm{GMm}}{\sqrt{2} r}+0=\frac{\mathrm{GM} m}{r}+\frac{1}{2} \mathrm{MV}^{2}$
<br/> <br/>or, $\frac{1}{2} \mathrm{MV}^{2}=\frac{\mathrm{GM} m}{r}\left[1-\frac{1}{\sqrt{2}}\right]$
<br/> <br/>$\Rightarrow \frac{1}{2} \mathrm{~V}^{2}=\frac{\mathrm{G} m}{r}\left[1-\frac{1}{\sqrt{2}}\right]$ or,
<br/> <br/>$\mathrm{V}=\sqrt{\frac{2 \mathrm{Gm}}{r}\left(1-\frac{1}{\sqrt{2}}\right)}$</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2016']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P1.db
title:
a-point-particle-of-mass-01-kg-is-executing-shm-of-amplitude-of-01-m-when-the-particle-passes-through-the-mean-position-its-kinetic-energy-is-8-1-0-3-

slug:
a-point-particle-of-mass-01-kg-is-executing-shm-of-amplitude-of-01-m-when-the-particle-passes-through-the-mean-position-its-kinetic-energy-is-8-1-0-3-21597

question:
<div class="question">A point particle of mass $0.1 \mathrm{~kg}$ is executing S.H.M. of amplitude of $0.1 \mathrm{~m}$. When the particle passes through the mean position, its kinetic energy is 8 $\times 10^{-3}$ Joule. Obtain the equation of motion of this particle if this initial phase of oscillation is $45^{\circ} .$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y=0.1 \sin \left(\pm 4 t+\frac{\pi}{4}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad y=0.2 \sin \left(\pm 4 t+\frac{\pi}{4}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y=0.1 \sin \left(\pm 2 t+\frac{\pi}{4}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y=0.2 \sin \left(\pm 2 t+\frac{\pi}{4}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=0.1 \sin \left(\pm 4 t+\frac{\pi}{4}\right)$</span> </div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch1_P1.db
title:
a-positive-charge-q-is-projected-in-magnetic-field-of-width-m-v-2-q-b-with-velocity-v-then-the-time-taken-by-charged-particle-to-emerge-from-the-magne

slug:
a-positive-charge-q-is-projected-in-magnetic-field-of-width-m-v-2-q-b-with-velocity-v-then-the-time-taken-by-charged-particle-to-emerge-from-the-magne-93250

question:
<div class="question">A positive charge <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> is projected in magnetic field of width <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>m</mi><mi>v</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mi>q</mi><mi>B</mi></mrow></mfrac></math> with velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>. Then, the time taken by charged particle to emerge from the magnetic field is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>m</mi><mrow><msqrt><mn>2</mn></msqrt><mi>q</mi><mi>B</mi></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>π</mi><mi>m</mi></mrow><mrow><mn>4</mn><mi>q</mi><mi>B</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>π</mi><mi>m</mi></mrow><mrow><mn>2</mn><mi>q</mi><mi>B</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>π</mi><mi>m</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mo>·</mo><mi>q</mi><mi>B</mi></mrow></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>π</mi><mi>m</mi></mrow><mrow><mn>4</mn><mi>q</mi><mi>B</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/78d4b281-ab47-4399-9786-9c285582d348-image.png" style="width: 300px; height: 294px;"/></p><p>Referring the diagram, </p><p>width of magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML"></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>v</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mi>q</mi><mi>B</mi></mrow></mfrac></math></p><p>Also we know that when a charge is projected in perpendicular magnetic field then radius of path is </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>v</mi></mrow><mrow><mi>q</mi><mi>B</mi></mrow></mfrac></math></p><p>Now in triangle OAB,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>θ</mi><mo>=</mo><mfrac><mi>w</mi><mi>R</mi></mfrac><mo>=</mo><mfrac><mfrac><mrow><mi>m</mi><mi>v</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mi>q</mi><mi>B</mi></mrow></mfrac><mfrac><mrow><mi>m</mi><mi>v</mi></mrow><mrow><mi>q</mi><mi>B</mi></mrow></mfrac></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>sin</mi><mi>θ</mi><mo>=</mo><mi>sin</mi><mn>45</mn><mo>°</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>sin</mi><mfrac><mi>π</mi><mn>4</mn></mfrac></math></p><p>Time period for angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi><mi>m</mi></mrow><mrow><mi>q</mi><mi>B</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Time by charged particle for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>in order to emerge from magnetic field is </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mfrac><mrow><mi>π</mi><mi>m</mi></mrow><mrow><mn>4</mn><mi>q</mi><mi>B</mi></mrow></mfrac></math>.</p></div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch1_P1.db
title:
a-projectile-a-is-thrown-at-an-angle-of-3-0-to-the-horizontal-from-point-p-at-the-same-time-another-projectile-b-is-thrown-with-velocity-v-2-upwards-f

slug:
a-projectile-a-is-thrown-at-an-angle-of-3-0-to-the-horizontal-from-point-p-at-the-same-time-another-projectile-b-is-thrown-with-velocity-v-2-upwards-f-99629

question:
<div class="question">A projectile A is thrown at an angle of $30^{\circ}$ to the horizontal from point P. At the same time, another projectile $\mathrm{B}$ is thrown with velocity $\mathrm{v}_{2}$ upwards from the point $Q$ vertically below the highest point. For $\mathrm{B}$ to collide with $\mathrm{A}, \frac{\mathrm{v}_{2}}{\mathrm{v}_{1}}$ should be
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/vF2xDXuYtbtwWbS7a7r-aDtGj8xqB_3Xt-aZ47JE_wU.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2}$</span> </div>

answer:
<div class="solution">This happen when vertical velocity of both are same.
<br/> <br/>$\therefore v_{2}=v_{1} \sin 30^{\circ} \text { or } \frac{v_{2}}{v_{1}}=\frac{1}{2}$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch1_P1.db
title:
a-projectile-can-have-the-same-range-r-for-two-angles-of-projection-if-t-1-and-t-2-be-the-times-of-flight-in-two-cases-then-what-is-the-product-of-two

slug:
a-projectile-can-have-the-same-range-r-for-two-angles-of-projection-if-t-1-and-t-2-be-the-times-of-flight-in-two-cases-then-what-is-the-product-of-two-34877

question:
<div class="question">A projectile can have the same range $\mathrm{R}$ for two angles of projection. If $t_{1}$ and $t_{2}$ be the times of flight in two cases, then what is the product of two times of flight?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{R}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{R}^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{l} / \mathrm{R}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{t}_{1} \mathrm{t}_{2} \propto 1 / \mathrm{R}^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{R}$</span> </div>

answer:
<div class="solution">$\mathrm{t}_{1}=\frac{2 \mathrm{u} \sin \theta}{\mathrm{g}}$ and
<br/>$\mathrm{t}_{2}=\frac{2 \mathrm{u} \sin (90-\theta)}{\mathrm{g}}=\frac{2 \mathrm{u} \cos \theta}{\mathrm{g}}$
<br/>$\begin{aligned} \therefore \mathrm{t}_{1} \mathrm{t}_{2} &amp;=\frac{4 \mathrm{u}^{2} \cos \theta \sin \theta}{\mathrm{g}^{2}}=\frac{2}{\mathrm{~g}}\left[\frac{\mathrm{u}^{2} \sin 2 \theta}{\mathrm{g}}\right] \\ &amp;=\frac{2}{\mathrm{~g}} \mathrm{R}, \end{aligned}$
<br/>where $\mathrm{R}$ is the range. 
<br/>Hence $t_{1} t_{2} \propto R$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch1_P1.db
title:
a-projectile-is-fired-with-a-velocity-u-making-an-angle-with-the-horizontal-what-is-the-magnitude-of-change-in-velocity-when-it-is-at-the-highest-poin

slug:
a-projectile-is-fired-with-a-velocity-u-making-an-angle-with-the-horizontal-what-is-the-magnitude-of-change-in-velocity-when-it-is-at-the-highest-poin-47206

question:
<div class="question">A projectile is fired with a velocity u making an angle $\theta$ with the horizontal. What is the magnitude of change in velocity when it is at the highest point -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$u \cos \theta$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{u}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{u} \sin \theta$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{u} \cos \theta-\mathrm{u}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{u} \sin \theta$</span> </div>

answer:
<div class="solution">Initially $\mathrm{u}=\cos \theta \hat{\mathrm{i}}+\mathrm{u} \sin \theta \hat{\mathrm{j}}$.
<br/> <br/>At highest point $\mathrm{v}=\mathrm{u} \cos \theta \hat{\mathrm{i}}$
<br/> <br/>$\therefore$ difference is $\mathrm{u} \sin \theta$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch1_P1.db
title:
a-projectile-is-given-an-initial-velocity-of-i-3-j-m-s-1-where-i-is-along-the-ground-and-j-is-along-the-vertical-then-the-equation-of-the-path-of-proj

slug:
a-projectile-is-given-an-initial-velocity-of-i-3-j-m-s-1-where-i-is-along-the-ground-and-j-is-along-the-vertical-then-the-equation-of-the-path-of-proj-47170

question:
<div class="question">A projectile is given an initial velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover></math> is along the ground and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math> is along the vertical. Then, the equation of the path of projectile is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo></math>Take, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open=""><mrow><mi>g</mi><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>x</mi><mo>-</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>x</mi><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>y</mi><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msqrt><mn>3</mn></msqrt></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>x</mi><mo>-</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Velocity of projectile at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mi>u</mi><mo>-</mo><mi>g</mi><mo> </mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mn>10</mn><mo> </mo><mi>t</mi><mover><mrow><mo> </mo><mi mathvariant="normal">j</mi></mrow><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo>  </mo><mi>r</mi><mo>=</mo><mo>∫</mo><mi>v</mi><mo> </mo><mi>d</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>∫</mo><mo> </mo><mfenced><mrow><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mn>10</mn><mo> </mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo> </mo><mi>d</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mn>10</mn><mfrac><msup><mi>t</mi><mn>2</mn></msup><mn>2</mn></mfrac><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mn>5</mn><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><msqrt><mn>3</mn></msqrt><mo> </mo><mi>t</mi><mo>-</mo><mn>5</mn><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mi>y</mi><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>=</mo><mi>t</mi><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><msqrt><mn>3</mn></msqrt><mo> </mo><mi>t</mi><mo>-</mo><mn>5</mn><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p>On comparing, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math>          ...(i)</p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mi>t</mi><mo>-</mo><mn>5</mn><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></math>          ...(ii)</p><p>From Eqs (i) and (ii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mi>x</mi><mo>-</mo><mn>5</mn><mo> </mo><msup><mi>x</mi><mn>2</mn></msup></math></p></div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch1_P1.db
title:
a-projectile-is-projected-at-3-0-from-horizontal-with-initial-velocity-40-ms-1-the-velocity-of-the-projectile-at-t-2-s-from-the-start-will-be-given-g-

slug:
a-projectile-is-projected-at-3-0-from-horizontal-with-initial-velocity-40-ms-1-the-velocity-of-the-projectile-at-t-2-s-from-the-start-will-be-given-g-39461

question:
<div class="question">A projectile is projected at $30^{\circ}$ from horizontal with initial velocity $40 \mathrm{~ms}^{-1}$. The velocity of the projectile at $\mathrm{t}=2 \mathrm{~s}$ from the start will be: (Given $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$20 \sqrt{3} \mathrm{~ms}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$40 \sqrt{3} \mathrm{~ms}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$20 \mathrm{~ms}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$20 \sqrt{3} \mathrm{~ms}^{-1}$</span> </div>

answer:
<div class="solution">Given,<br/>Initial velocity of projectile, $u=40 \mathrm{~m} / \mathrm{s}$<br/>Angle, $\theta=30^{\circ}$<br/>Time of flight<br/>$\mathrm{T}=\frac{2 \mathrm{u} \sin \theta}{\mathrm{g}}=\frac{2 \times 40 \times 1}{10 \times 2}=45\left(\because \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right)$<br/>It means projectile is at maximum height at $t=$ $2 \mathrm{~s}$. At maximum height vertical component of velocity is zero.<br/>$\begin{aligned}<br/>&amp; \text { Velocity at } \mathrm{t}=2 \mathrm{~s}=\mathrm{V}_{\mathrm{x}}=\mathrm{u} \cos \theta=40 \cos 30^{\circ} \\<br/>&amp; =20 \sqrt{3} \mathrm{~ms}^{-1} \text {. } \\<br/>&amp;<br/>\end{aligned}$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P1.db
title:
a-projectile-is-projected-with-the-velocity-of-3-i-4-j-m-s-the-horizontal-range-of-the-projectile-will-be

slug:
a-projectile-is-projected-with-the-velocity-of-3-i-4-j-m-s-the-horizontal-range-of-the-projectile-will-be-36322

question:
<div class="question">A projectile is projected with the velocity of $(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$. The horizontal range of the projectile will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$1.2 \mathrm{~m}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$2.4 \mathrm{~m}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$3.6 \mathrm{~m}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$4.5 \mathrm{~m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$2.4 \mathrm{~m}$
<br/></span> </div>

answer:
<div class="solution">Given, $\mathbf{v}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}$
<br/>$$
<br/>\therefore \quad v=|\mathrm{v}|=\sqrt{3^2+4^2} \Rightarrow v=5 \mathrm{~m} / \mathrm{s}
<br/>$$
<br/>From figure,
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/S68DyZBu4WVIyvgm-kVZTbqOsJd6ueHWTqIkKHSEDW4.original.fullsize.png"/><br/> <br/>$\sin \theta=\frac{4}{5}$ and $\cos \theta=\frac{3}{5}$
<br/>$\because \quad R=\frac{v^2 \sin 2 \theta}{g}$
<br/>$=\frac{v^2 \cdot 2 \sin \theta \cdot \cos \theta}{g}$
<br/>$=\frac{5 \times 5 \times 2 \times \frac{4}{5} \times \frac{3}{5}}{10}=24 \mathrm{~m}$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch1_P1.db
title:
a-projectile-projected-at-an-angle-3-0-from-the-horizontal-has-a-range-r-if-the-angle-of-projection-at-the-same-initial-velocity-be-6-0-then-the-range

slug:
a-projectile-projected-at-an-angle-3-0-from-the-horizontal-has-a-range-r-if-the-angle-of-projection-at-the-same-initial-velocity-be-6-0-then-the-range-24576

question:
<div class="question">A projectile projected at an angle $30^{\circ}$ from the horizontal has a range $\mathrm{R}$. If the angle of projection at the same initial velocity be $60^{\circ}$, then the range will be-</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{R}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{R} / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{R}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{R}^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{R}$</span> </div>

answer:
<div class="solution">If sum of angle of projection $=90^{\circ}$ for given speed then range for that angle of projection is same.</div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch1_P1.db
title:
a-proton-and-an-particle-are-accelerated-from-rest-by-2-v-and-4-v-potentials-respectively-the-ratio-of-their-debroglie-wavelength-is-1

slug:
a-proton-and-an-particle-are-accelerated-from-rest-by-2-v-and-4-v-potentials-respectively-the-ratio-of-their-debroglie-wavelength-is-1-70167

question:
<div class="question">A proton and an $\alpha$-particle are accelerated from rest by $2 \mathrm{~V}$ and $4 \mathrm{~V}$ potentials, respectively. The ratio of their de-Broglie wavelength is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>4 : 1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>2 : 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>8 : 1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>16 : 1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>4 : 1</span> </div>

answer:
<div class="solution">De-Broglie wavelength,<br/>$\begin{aligned}<br/>&amp; \lambda=\frac{\mathrm{h}}{\mathrm{mv}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mK}}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mq} \Delta \mathrm{V}}} \\<br/>&amp; \therefore \frac{\lambda_\alpha}{\lambda_{\mathrm{p}}}=\sqrt{\frac{2 \mathrm{~m}_{\mathrm{p}} \mathrm{V}_{\mathrm{p}} \mathrm{q}_{\mathrm{p}}}{2 \mathrm{~m}_\alpha \mathrm{V}_\alpha \mathrm{q}_\alpha}} \\<br/>&amp; \because \mathrm{V}_{\mathrm{p}}=2 \mathrm{~V} \text { and } \mathrm{V}_\alpha=4 \mathrm{~V} \text { (given) } \\<br/>&amp; \mathrm{m}_\alpha=4 \mathrm{~m}_{\mathrm{p}} \text { and } \mathrm{q}_\alpha=2 \mathrm{q}_{\mathrm{p}}<br/>\end{aligned}$<br/>$\therefore \frac{\lambda_\alpha}{\lambda_p}=\sqrt{\frac{1 \times 2 \times 1}{4 \times 4 \times 2}}=\frac{1}{4} \Rightarrow \lambda_p: \lambda_\alpha=4: 1$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch1_P1.db
title:
a-proton-is-revolving-on-a-circular-path-of-radius-2-mm-with-frequency-10-hz-magnetic-dipole-moment-associated-with-proton-is

slug:
a-proton-is-revolving-on-a-circular-path-of-radius-2-mm-with-frequency-10-hz-magnetic-dipole-moment-associated-with-proton-is-59001

question:
<div class="question">A proton is revolving on a circular path of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>mm</mi></math> with frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi>Hz</mi></math>. Magnetic dipole moment associated with proton is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>24</mn></mrow></msup><mo> </mo><mi mathvariant="normal">A</mi><mo>-</mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>24</mn></mrow></msup><mo> </mo><mi mathvariant="normal">A</mi><mo>-</mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>20</mn></mrow></msup><mo> </mo><mi mathvariant="normal">A</mi><mo>-</mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>20</mn></mrow></msup><mo> </mo><mi mathvariant="normal">A</mi><mo>-</mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>24</mn></mrow></msup><mo> </mo><mi mathvariant="normal">A</mi><mo>-</mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Given, charge on proton, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>e</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>19</mn></mrow></msup><mo> </mo><mi mathvariant="normal">C</mi></math></p><p>Radius, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>  </mo><mi>r</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>mm</mi><mo>=</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi>cm</mi></math></p><p>Frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>10</mn><mo> </mo><mi>Hz</mi></math></p><p>Current associated by proton is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mfrac><mi>q</mi><mi>t</mi></mfrac></math>                       <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>T</mi><mo>→</mo></math> time period <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>q</mi><mi>f</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>  </mo><mfenced close="]" open="["><mrow><mo>∵</mo><mi>T</mi><mo>=</mo><mfrac><mn>1</mn><mi>f</mi></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>19</mn></mrow></msup><mo>×</mo><mn>10</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>18</mn></mrow></msup><mo> </mo><mi mathvariant="normal">A</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Magnetic dipole moment, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mi>I</mi><mi>A</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>I</mi><mo>·</mo><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>18</mn></mrow></msup><mo>×</mo><mn>3</mn><mo>.</mo><mn>14</mn><mo>×</mo><msup><mfenced><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfenced><mn>2</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>24</mn></mrow></msup><mo> </mo><mi mathvariant="normal">A</mi><mo>-</mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math></p></div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
title:
a-proton-moving-with-a-velocity-3-1-0-5-m-s-enters-a-magnetic-field-of-03-tesla-at-an-angle-of-3-0-with-the-field-the-radius-of-curvature-of-its-path-

slug:
a-proton-moving-with-a-velocity-3-1-0-5-m-s-enters-a-magnetic-field-of-03-tesla-at-an-angle-of-3-0-with-the-field-the-radius-of-curvature-of-its-path-93899

question:
<div class="question">A proton moving with a velocity $3 \times 10^{5} \mathrm{~m} / \mathrm{s}$ enters a magnetic field of $0.3$ tesla at an angle of $30^{\circ}$ with the field. The radius of curvature of its path will be $(\mathrm{e} / \mathrm{m}$ for proton $=10^{8} \mathrm{C} / \mathrm{kg}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.5 \mathrm{~cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.02 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.25 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.5 \mathrm{~cm}$</span> </div>

answer:
<div class="solution">$r=\frac{m v \sin \theta}{\mathrm{B} e}=\frac{3 \times 10^{5} \sin 30^{\circ}}{0.3 \times 10^{8}}$
<br/> <br/>$\frac{3 \times 10^{5} \times \frac{1}{2}}{3 \times 10^{7}}=0.5 \times 10^{-2}$
<br/> <br/>$m=0.5 \mathrm{~cm}$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-pulley-of-radius-2-m-is-rotated-about-its-axis-by-a-force-20-t-5-t-2-newton-where-t-is-measured-in-seconds-applied-tangentially-if-the-moment-of-ine

slug:
a-pulley-of-radius-2-m-is-rotated-about-its-axis-by-a-force-20-t-5-t-2-newton-where-t-is-measured-in-seconds-applied-tangentially-if-the-moment-of-ine-75618

question:
<div class="question">A pulley of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math> is rotated about its axis by a force <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>20</mn><mi>t</mi><mo>-</mo><mn>5</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></math> newton (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup><mo>,</mo></math> the number of rotation made by the pulley before its direction of motion is reversed is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> but less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> but less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> but less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p>Torque <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>r</mi><mo>×</mo><mi>F</mi><mo>=</mo><mi>l</mi><mi>α</mi></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mn>20</mn><mi>t</mi><mo>-</mo><mn>5</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>10</mn><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>4</mn><mi>t</mi><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></math></p><p>But <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>ω</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>,</mo></math> so</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>ω</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn>4</mn><mi>t</mi><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>ω</mi><mo>=</mo><mfenced><mrow><mn>4</mn><mi>t</mi><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>t</mi></math></p><p>Integrating <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mfrac><msup><mi>t</mi><mn>3</mn></msup><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> will be zero at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>6</mn><mo> </mo><mi mathvariant="normal">s</mi></math></p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>θ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mfrac><msup><mi>t</mi><mn>3</mn></msup><mn>3</mn></mfrac></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>θ</mi><mo>=</mo><mfenced><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mfrac><msup><mi>t</mi><mn>3</mn></msup><mn>3</mn></mfrac></mrow></mfenced><mi>d</mi><mi>t</mi></math></p><p>Again, integrating both sides<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>t</mi><mn>3</mn></msup></mrow><mn>3</mn></mfrac><mo>-</mo><mfrac><msup><mi>t</mi><mn>4</mn></msup><mn>12</mn></mfrac></math></p><p>since. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>6</mn><mo> </mo><mi mathvariant="normal">s</mi></math></p><p>so, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mo>×</mo><msup><mn>6</mn><mn>3</mn></msup></mrow><mn>3</mn></mfrac><mo>-</mo><mfrac><msup><mn>6</mn><mn>4</mn></msup><mn>12</mn></mfrac><mo>=</mo><mn>36</mn></math></p><p>Number of turns, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mi>θ</mi><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mo>=</mo><mfrac><mn>36</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mo>=</mo><mn>5</mn><mo>.</mo><mn>73</mn></math></p></div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch1_P1.db
title:
a-radioactive-element-x-converts-into-another-stable-element-y-half-life-of-x-is-2-hrs-initially-only-x-is-present-after-time-t-the-ratio-of-atoms-of-

slug:
a-radioactive-element-x-converts-into-another-stable-element-y-half-life-of-x-is-2-hrs-initially-only-x-is-present-after-time-t-the-ratio-of-atoms-of-31478

question:
<div class="question">A radioactive element X converts into another stable element Y. Half life of $\mathrm{X}$ is $2 \mathrm{hrs}$. Initially only $\mathrm{X}$ is present. After time $\mathrm{t},$ the ratio of atoms of $X$ and $Y$ is found to be $1: 4,$ then $t$ in hours is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">between 4 and 6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">between 4 and 6</span> </div>

answer:
<div class="solution">Let $\mathrm{N}_{0}$ be the number of atoms of $\mathrm{X}$ at time $\mathrm{t}=0$
<br/> <br/>Then at $\mathrm{t}=4$ hrs (two half lives)
<br/> <br/>$\mathrm{N}_{\mathrm{x}}=\frac{\mathrm{N}_{0}}{4}$ and $\mathrm{N}_{\mathrm{y}}=\frac{3 \mathrm{~N}_{0}}{4}$
<br/> <br/>$\therefore \mathrm{N}_{\mathrm{x}} / \mathrm{N}_{\mathrm{y}}=1 / 3$
<br/> <br/>and at $\mathrm{t}=6 \mathrm{hrs}$ (three half lives)
<br/> <br/>$\mathrm{N}_{\mathrm{x}}=\frac{\mathrm{N}_{0}}{8}$ and $\mathrm{N}_{\mathrm{y}}=\frac{7 \mathrm{~N}_{0}}{8}$
<br/> <br/>or $\frac{\mathrm{N}_{\mathrm{x}}}{\mathrm{N}_{\mathrm{y}}}=\frac{1}{7}$
<br/> <br/>The given ratio $\frac{1}{4}$ lies between $\frac{1}{3}$ and $\frac{1}{7}$ Therefore, t lies between $4 \mathrm{hrs}$ and $6 \mathrm{hrs}$.</div>


tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch1_P1.db
title:
a-radioactive-isotope-having-a-half-life-period-of-3-days-was-received-after-12-days-if-3-g-of-the-isotope-is-left-in-the-container-what-would-be-the--1

slug:
a-radioactive-isotope-having-a-half-life-period-of-3-days-was-received-after-12-days-if-3-g-of-the-isotope-is-left-in-the-container-what-would-be-the-1-50396

question:
<div class="question">A radioactive isotope having a half - life period of 3 days was received after 12 days. If $3 \mathrm{~g}$ of the isotope is left in the container, what would be the initial mass of the isotope?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$12 \mathrm{~g}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$36 \mathrm{~g}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$48 \mathrm{~g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$24 \mathrm{~g}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$48 \mathrm{~g}$</span> </div>

answer:
<div class="solution">Given $t_{1 / 2}=3$ Total time $\mathrm{T}=12$
<br/> <br/>No. of half lives $(n)=\frac{12}{3}=4$ $\left(\frac{1}{2}\right)^{n}=\frac{\mathrm{N}}{\mathrm{N}_{\mathrm{o}}} 
<br/> <br/>\therefore\left(\frac{1}{2}\right)^{4}=\frac{3}{\mathrm{~N}}$
<br/> <br/>$\frac{3}{\mathrm{~N}}=\frac{1}{16}$
<br/> <br/>$N=48$</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch1_P1.db
title:
a-radioactive-material-is-reduced-to-18-of-its-original-amount-in-3-days-if-8-1-0-3-kg-of-the-material-is-left-after-5-days-the-initial-amount-of-the-

slug:
a-radioactive-material-is-reduced-to-18-of-its-original-amount-in-3-days-if-8-1-0-3-kg-of-the-material-is-left-after-5-days-the-initial-amount-of-the-52547

question:
<div class="question">A radio-active material is reduced to $1 / 8$ of its original amount in 3 days. If $8 \times 10^{-3} \mathrm{~kg}$ of the material is left after 5 days. The initial amount of the material is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$700 \mathrm{gm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$900 \mathrm{gm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$475 \mathrm{gm}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$256 \mathrm{gm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$256 \mathrm{gm}$</span> </div>

answer:
<div class="solution">$\mathrm{N}=\mathrm{N}_0\left(\frac{1}{2}\right)^{\mathrm{n}}$<br/>$\mathrm{N}=\frac{\mathrm{N}_0}{8}$<br/>$\frac{\mathrm{N}_0}{8}=\mathrm{N}_0\left(\frac{1}{2}\right)^{\mathrm{n}} \Rightarrow\left(\frac{1}{2}\right)^3=\left(\frac{1}{2}\right)^{\mathrm{n}}$<br/>$\mathrm{n}=3$<br/>3 half lives $=3$ days<br/>1 half life $=1$ day<br/>5 days $=5$ half life<br/>$\mathrm{N}=\mathrm{N}_0\left(\frac{1}{2}\right)^{\mathrm{n}} \Rightarrow 8 \times 10^{-3}=\mathrm{N}_0\left(\frac{1}{2}\right)^5$<br/>$\Rightarrow \quad N_0=2^5 \times 8 \times 10^{-3}=256 \mathrm{gm}$</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch1_P1.db
title:
a-radioactive-sample-of-half-life-23-1-days-is-disintegrating-continuously-the-percentage-decay-of-the-sample-between-the-15-th-and-16-th-day-will-be-

slug:
a-radioactive-sample-of-half-life-23-1-days-is-disintegrating-continuously-the-percentage-decay-of-the-sample-between-the-15-th-and-16-th-day-will-be-58057

question:
<div class="question">A radioactive sample of half life <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>23</mn><mo>.</mo><mn>1</mn><mo> </mo><mi>days</mi></math> is disintegrating continuously. The percentage decay of the sample between the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>15</mn><mi>th</mi></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>16</mn><mi>th</mi></msup></math> day will be [Take, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mn>0</mn><mo>.</mo><mn>03</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>.</mo><mn>03</mn></math>]</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><mo>%</mo></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mo>%</mo></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>9</mn><mo> </mo><mo>%</mo></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn><mo> </mo><mo>%</mo></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>9</mn><mo> </mo><mo>%</mo></math></span> </div>

answer:
<div class="solution"><p>Given, half life <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></mfenced><mo>=</mo><mn>23</mn><mo>.</mo><mn>1</mn></math> days</p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>0</mn></msub></math> be the initial amount, then active nuclei at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>15</mn></math> days</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>1</mn></msub><mo>=</mo><msub><mi>N</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>15</mn><mi>λ</mi></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>Active nuclei at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>16</mn></math> days</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>2</mn></msub><mo>=</mo><msub><mi>N</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>16</mn><mi>λ</mi></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>Disintegration constant,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>693</mn></mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></mfrac><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>693</mn></mrow><mrow><mn>23</mn><mo>.</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>03</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>%</mo></math> decay in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mi>th</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mi>th</mi></math> days</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mn>1</mn></msub><mo>-</mo><msub><mi>N</mi><mn>2</mn></msub></mrow><msub><mi>N</mi><mn>1</mn></msub></mfrac><mo>×</mo><mn>100</mn><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>15</mn><mi>λ</mi></mrow></msup><mo>-</mo><msub><mi>N</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>16</mn><mi>λ</mi></mrow></msup></mrow><mrow><msub><mi>N</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>15</mn><mi>λ</mi></mrow></msup></mrow></mfrac><mo>×</mo><mn>100</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>λ</mi></mrow></msup></mrow></mfenced><mo>×</mo><mn>100</mn><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>e</mi><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>03</mn></mrow></msup></mrow></mfenced><mo>×</mo><mn>100</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><msup><mi>e</mi><mrow><mn>0</mn><mo>.</mo><mn>03</mn></mrow></msup></mfrac></mrow></mfenced><mo>×</mo><mn>100</mn><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>.</mo><mn>03</mn></mrow></mfrac></mrow></mfenced><mo>×</mo><mn>100</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>9</mn><mo> </mo><mo>%</mo></math></p></div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2016']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P1.db
title:
a-ray-of-light-coming-from-the-point-12-is-reflected-at-a-point-a-on-the-x-axis-and-then-passes-through-the-point-5-3-the-coordinates-of-the-point-a-i

slug:
a-ray-of-light-coming-from-the-point-12-is-reflected-at-a-point-a-on-the-x-axis-and-then-passes-through-the-point-5-3-the-coordinates-of-the-point-a-i-50690

question:
<div class="question">A ray of light coming from the point (1,2) is reflected at a point A on the $x$ -axis and then passes through the point $(5,3) .$ The co-ordinates of the point $A$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\frac{13}{5}, 0\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{5}{13}, 0\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(-7,0)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{13}{5}, 0\right)$</span> </div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch1_P1.db
title:
a-ray-of-light-is-incident-at-the-glasswater-interface-at-an-angle-i-it-emerges-finally-parallel-to-the-surface-of-water-then-the-value-of-g-would-be

slug:
a-ray-of-light-is-incident-at-the-glasswater-interface-at-an-angle-i-it-emerges-finally-parallel-to-the-surface-of-water-then-the-value-of-g-would-be-22403

question:
<div class="question">A ray of light is incident at the glass-water interface at an angle i, it emerges finally parallel to the surface of water, then the value of $\mu_{\mathrm{g}}$ would be
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/w84qX6WRFSY6p-nNmtWiyMfUP0Y6b3wXeEmORQ98xv8.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(4 / 3) \sin i$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 / \sin i$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 / 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / \sin i$</span> </div>

answer:
<div class="solution">$\mu_{g} \sin i=\mu_{\text {air }} \sin 90^{\circ} \Rightarrow \mu_{g}=\frac{1}{\sin i}$</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">49: 1</span> </div>
source_db: MarksBatch1_P1.db
title:
a-ray-of-light-of-intensity-i-is-incident-on-a-parallel-glass-slab-at-point-a-as-shown-in-diagram-it-undergoes-partial-reflection-and-refraction-at-ea-1

slug:
a-ray-of-light-of-intensity-i-is-incident-on-a-parallel-glass-slab-at-point-a-as-shown-in-diagram-it-undergoes-partial-reflection-and-refraction-at-ea-1-94534

question:
<div class="question">A ray of light of intensity I is incident on a parallel glass slab at point $A$ as shown in diagram. It undergoes partial reflection and refraction. At each reflection, $25 \%$ of incident energy is reflected. The rays $\mathrm{AB}$ and $\mathrm{A}^{\prime} \mathrm{B}^{\prime}$ undergo interference. The ratio of $\mathrm{I}_{\max }$ and $\mathrm{I}_{\min }$ is :
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/BeHYcB4nAXifb9OiyggZJVUqMftkEO7FaHfVoh3spYw.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">49: 1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7: 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4: 1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8: 1</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/IYtubaSAS9B_I-_VHoJuIo93YIPa5VI5ha3u9RLc9Eo.original.fullsize.png"/><br/> <br/> <br/>From figure $\mathrm{I}_{1}=\frac{\mathrm{I}}{4}$ and $\mathrm{I}_{2}=\frac{9 \mathrm{I}}{64}$
<br/> <br/>$\begin{aligned} \text { By using } \frac{\mathrm{I}_{\max }}{\mathrm{I}_{\min }} &amp;=\left(\frac{\sqrt{\frac{\mathrm{I}_{2}}{\mathrm{I}_{1}}}+1}{\sqrt{\frac{\mathrm{I}_{2}}{\mathrm{I}_{1}}}-1}\right)^{2} \\ &amp;=\left(\frac{\sqrt{\frac{9}{16}}+1}{\sqrt{\frac{9}{16}}-1}\right)^{2}=\frac{49}{1} \end{aligned}$</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch1_P1.db
title:
a-ray-parallel-to-principal-axis-is-incident-at-3-0-from-normal-on-concave-mirror-having-radius-of-curvature-r-the-point-on-principal-axis-where-rays-

slug:
a-ray-parallel-to-principal-axis-is-incident-at-3-0-from-normal-on-concave-mirror-having-radius-of-curvature-r-the-point-on-principal-axis-where-rays-51163

question:
<div class="question">A ray parallel to principal axis is incident at $30^{\circ}$ from normal on concave mirror having radius of curvature $R$. The point on principal axis where rays are focussed is $Q$ such that $\mathrm{PQ}$ is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rueiuBD3_u-ZkcDC2Wdf5GeygQBjLNAolOzYsaqtes0.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{R}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{R}{\sqrt{3}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2 \sqrt{R}-R}{\sqrt{2}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$R\left(1-\frac{1}{\sqrt{3}}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$R\left(1-\frac{1}{\sqrt{3}}\right)$</span> </div>

answer:
<div class="solution">From similar triangles,
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/XHNSXpn6kBt78iTdzYBxZhp-Uyp1yRN1TqalGBhb0xg.original.fullsize.png"/><br/> <br/> <br/>$$
<br/> <br/>\frac{Q C}{\sin 30^{\circ}}=\frac{R}{\sin 120^{\circ}}
<br/> <br/>$$
<br/> <br/>or $\quad Q C=R \times \frac{\sin 30^{\circ}}{\sin 120^{\circ}}=\frac{R}{\sqrt{3}}$
<br/> <br/>Thus $P Q=P C-Q C=R-\frac{R}{\sqrt{3}}=R\left(1-\frac{1}{\sqrt{3}}\right)$</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch1_P1.db
title:
a-rectangular-block-of-mass-m-and-area-of-crosssection-a-floats-in-a-liquid-of-density-if-it-is-given-a-vertical-displacement-from-equilibrium-it-unde

slug:
a-rectangular-block-of-mass-m-and-area-of-crosssection-a-floats-in-a-liquid-of-density-if-it-is-given-a-vertical-displacement-from-equilibrium-it-unde-87166

question:
<div class="question">A rectangular block of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and area of cross-section <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> floats in a liquid of density <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math>. If it is given a vertical displacement from equilibrium, it undergoes oscillation with a time period <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>∝</mo><msqrt><mi>ρ</mi></msqrt></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>∝</mo><mfrac><mn>1</mn><msqrt><mi>A</mi></msqrt></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>∝</mo><mfrac><mn>1</mn><mi>ρ</mi></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>∝</mo><mfrac><mn>1</mn><msqrt><mi>m</mi></msqrt></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>∝</mo><mfrac><mn>1</mn><msqrt><mi>A</mi></msqrt></mfrac></math></span> </div>

answer:
<div class="solution"><p>Let block is displaced through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mi mathvariant="normal">m</mi></math>, then weight of displaced water or upthrust (upwards)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>A</mi><mi>x</mi><mi>ρ</mi><mi>g</mi></math></p><p>Where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is area of a cross-section of the block and $\rho$ is density. This must be equal to force <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mi>m</mi><mi>a</mi></mrow></mfenced></math> applied where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is mass of the block and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is acceleration.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>m</mi><mi>a</mi><mo>=</mo><mo>-</mo><mi>A</mi><mi>x</mi><mi>ρ</mi><mi>g</mi></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>A</mi><mi>ρ</mi><mi>g</mi></mrow><mi>m</mi></mfrac><mi>x</mi><mo>=</mo><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mi>x</mi></math></p><p>This is equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>SHM</mi></math>.</p><p>The time period of oscillation,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>ω</mi></mfrac><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mfrac><mi>m</mi><mrow><mi>A</mi><mi>ρ</mi><mi>g</mi></mrow></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>T</mi><mo>∝</mo><mfrac><mn>1</mn><msqrt><mi>A</mi></msqrt></mfrac></math></p></div>


title: a-reduction-b-chcl-3-koh-c-reduction-n-methylaniline-a-is
slug: a-reduction-b-chcl-3-koh-c-reduction-n-methylaniline-a-is-90311
tags: ['Chemistry', 'Amines', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
question:
<div class="question">$[\mathrm{A}] \stackrel{\text { reduction }}{\longrightarrow}[\mathrm{B}] \stackrel{\mathrm{CHCl}_{3}+\mathrm{KOH}}{\longrightarrow}[\mathrm{C}] \stackrel{\text { reduction }}{\longrightarrow} \mathrm{N}-$ Methylaniline, A is?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Formaldehyde</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Trichloromethane</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Nitrobenzene</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Toluene</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Nitrobenzene</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PDsYarnqvnc295Z_m5Jv-KN_v0rhAajUG7c3K8XplGY.original.fullsize.png"/></div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-reservoir-is-at-827-c-and-carnots-engine-takes-a-thousand-kilo-calorie-of-heat-from-it-and-exhausts-it-to-a-sink-at-27-c-what-is-the-amount-of-work-

slug:
a-reservoir-is-at-827-c-and-carnots-engine-takes-a-thousand-kilo-calorie-of-heat-from-it-and-exhausts-it-to-a-sink-at-27-c-what-is-the-amount-of-work-58301

question:
<div class="question">A reservoir is at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>827</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> and Carnot's engine takes a thousand <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>kilo</mi><mi>calorie</mi></math> of heat from it and exhausts it to a sink at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math>. What is the amount of work and the efficiency of the engine?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>7</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi>cal</mi><mo>,</mo><mo> </mo><mn>70</mn><mo>.</mo><mn>70</mn><mo> </mo><mo>%</mo></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>27</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi>cal</mi><mo>,</mo><mo> </mo><mn>72</mn><mo>.</mo><mn>72</mn><mo> </mo><mo>%</mo></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>70</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi>cal</mi><mo>,</mo><mo> </mo><mn>80</mn><mo>.</mo><mn>70</mn><mo> </mo><mo>%</mo></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>70</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi>cal</mi><mo>,</mo><mo> </mo><mn>70</mn><mo>.</mo><mn>70</mn><mo> </mo><mo>%</mo></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>27</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi>cal</mi><mo>,</mo><mo> </mo><mn>72</mn><mo>.</mo><mn>72</mn><mo> </mo><mo>%</mo></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mn>1</mn></msub><mo>=</mo><msup><mn>10</mn><mn>6</mn></msup><mo> </mo><mi>cal</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub><mo>=</mo><mn>827</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi><mo>=</mo><mo>(</mo><mn>827</mn><mo>+</mo><mn>273</mn><mo>)</mo><mo>=</mo><mn>1100</mn><mo> </mo><mi mathvariant="normal">K</mi></math>,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub><mo>=</mo><mn>27</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi><mo>=</mo><mo>(</mo><mn>27</mn><mo>+</mo><mn>273</mn><mo>)</mo><mo>=</mo><mn>300</mn><mo> </mo><mi mathvariant="normal">K</mi></math>.</p><p>As, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>  </mo><mfrac><msub><mi>Q</mi><mn>1</mn></msub><msub><mi>T</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>Q</mi><mn>2</mn></msub><msub><mi>T</mi><mn>2</mn></msub></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo>  </mo><msub><mi>Q</mi><mn>2</mn></msub><mo>=</mo><mfrac><msub><mi>T</mi><mn>2</mn></msub><msub><mi>T</mi><mn>1</mn></msub></mfrac><msub><mi>Q</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>300</mn><mn>1100</mn></mfrac><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>72</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi>cal</mi></math><br/>work done, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><msub><mi>Q</mi><mn>1</mn></msub><mo>-</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>=</mo><msup><mn>10</mn><mn>6</mn></msup><mo>-</mo><mn>2</mn><mo>.</mo><mn>72</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo>=</mo><mn>7</mn><mo>.</mo><mn>27</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi>cal</mi></math></p><p>The efficiency of the engine,<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>T</mi><mn>2</mn></msub><msub><mi>T</mi><mn>1</mn></msub></mfrac></mrow></mfenced><mo>×</mo><mn>100</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>η</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>300</mn><mn>1100</mn></mfrac></mrow></mfenced><mo>×</mo><mn>100</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>72</mn><mo>.</mo><mn>72</mn><mo> </mo><mo>%</mo></math></p></div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch1_P1.db
title:
a-resistance-r-and-inductance-l-and-a-capacitor-c-all-are-connected-in-series-with-an-ac-supply-the-resistance-of-r-is-16-ohm-and-for-a-given-frequenc

slug:
a-resistance-r-and-inductance-l-and-a-capacitor-c-all-are-connected-in-series-with-an-ac-supply-the-resistance-of-r-is-16-ohm-and-for-a-given-frequenc-49901

question:
<div class="question">A resistance $\mathrm{R}$ and inductance $\mathrm{L}$ and a capacitor $\mathrm{C}$ all are connected in series with an AC supply. The resistance of $\mathrm{R}$ is 16 ohm and for a given frequency, the inductive reactance of $L$ is 24 ohm and capacitive reactance of $\mathrm{C}$ is $12 \mathrm{ohm} .$ If the current in the circuit is 5 amp., find the potential difference across $\mathrm{R}, \mathrm{L}$ and $\mathrm{C}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">30,20,50 volt</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">40,100,60 volt</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">70,110,60 volt</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">80,120,60 volt</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">80,120,60 volt</span> </div>

answer:
<div class="solution">$\mathrm{V}_{\mathrm{R}}=\mathrm{i} \mathrm{R}=5 \times 16=80 \mathrm{Volt}$<br/>$$<br/>\begin{array}{l}<br/>\mathrm{V}_{\mathrm{L}}=\mathrm{i} \times(\omega \mathrm{L})=5 \times 24=120 \mathrm{Volt} \\<br/>\mathrm{V}_{\mathrm{C}}=\mathrm{i} \times(\mathrm{l} / \omega \mathrm{C})=5 \times 12=60 \mathrm{Volt}<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Alternating Current', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-resistor-of-resistance-r-capacitor-of-capacitance-c-and-inductor-of-inductance-l-are-connected-in-parallel-to-ac-power-source-of-voltage-0-sin-t-the-1

slug:
a-resistor-of-resistance-r-capacitor-of-capacitance-c-and-inductor-of-inductance-l-are-connected-in-parallel-to-ac-power-source-of-voltage-0-sin-t-the-1-57255

question:
<div class="question">A resistor of resistance $R$, capacitor of capacitance $C$ and inductor of inductance $L$ are connected in parallel to AC power source of voltage $\varepsilon_{0} \sin \omega t$. The maximum current through the resistance is half of the maximum current through the power source. Then value of $\mathrm{R}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{3}}{\left|\omega C-\frac{1}{\omega L}\right|}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{3}\left|\frac{1}{\omega C}-\omega L\right|$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{5}\left|\frac{1}{\omega C}-\omega L\right|$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3}}{\left|\omega C-\frac{1}{\omega L}\right|}$</span> </div>

answer:
<div class="solution">$\frac{i_{R_{0}}}{\sqrt{\left(i_{R_{0}}\right)^{2}+\left(i_{c_{0}}-i_{L_{0}}\right)^{2}}}=\frac{1}{2}$
<br/> <br/>$\Rightarrow \frac{\varepsilon_{0} / R}{\sqrt{\left(\varepsilon_{0} / R\right)^{2}+\left(\varepsilon_{0} \omega C-\frac{\varepsilon_{0}}{\omega L}\right)^{2}}}=\frac{1}{2}$
<br/> <br/>$\Rightarrow R=\frac{\sqrt{3}}{\left(\omega C-\frac{1}{\omega L}\right)}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Rrn4TWflryKKMeTqlt_-v_Kj9S0PR4CKBdOoea_aOMY.original.fullsize.png"/></div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch1_P1.db
title:
a-resistor-of-resistance-r-capacitor-of-capacitance-c-and-inductor-of-inductance-l-are-connected-in-parallel-to-ac-power-source-of-voltage-0-sin-t-the

slug:
a-resistor-of-resistance-r-capacitor-of-capacitance-c-and-inductor-of-inductance-l-are-connected-in-parallel-to-ac-power-source-of-voltage-0-sin-t-the-51891

question:
<div class="question">A resistor of resistance $R$, capacitor of capacitance $C$ and inductor of inductance $L$ are connected in parallel to AC power source of voltage $\varepsilon_{0} \sin \omega t$. The maximum current through the resistance is half of the maximum current through the power source. Then value of $\mathrm{R}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{3}}{\left|\omega C-\frac{1}{\omega L}\right|}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{3}\left|\frac{1}{\omega C}-\omega L\right|$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{5}\left|\frac{1}{\omega C}-\omega L\right|$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3}}{\left|\omega C-\frac{1}{\omega L}\right|}$</span> </div>

answer:
<div class="solution">$\frac{i_{R_{0}}}{\sqrt{\left(i_{R_{0}}\right)^{2}+\left(i_{c_{0}}-i_{L_{0}}\right)^{2}}}=\frac{1}{2}$
<br/> <br/>$\Rightarrow \frac{\varepsilon_{0} / R}{\sqrt{\left(\varepsilon_{0} / R\right)^{2}+\left(\varepsilon_{0} \omega C-\frac{\varepsilon_{0}}{\omega L}\right)^{2}}}=\frac{1}{2}$
<br/> <br/>$\Rightarrow R=\frac{\sqrt{3}}{\left(\omega C-\frac{1}{\omega L}\right)}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Rrn4TWflryKKMeTqlt_-v_Kj9S0PR4CKBdOoea_aOMY.original.fullsize.png"/></div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-resistor-r-an-inductor-l-a-capacitor-c-and-voltmeters-v-1-v-2-and-v-3-are-connected-to-an-oscillator-in-the-circuit-as-shown-in-the-adjoining-diagra

slug:
a-resistor-r-an-inductor-l-a-capacitor-c-and-voltmeters-v-1-v-2-and-v-3-are-connected-to-an-oscillator-in-the-circuit-as-shown-in-the-adjoining-diagra-88188

question:
<div class="question">A resistor <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>, an inductor <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>, a capacitor <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> and voltmeters <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>1</mn></msub><mo>,</mo><msub><mi>V</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>3</mn></msub></math> are connected to an oscillator in the circuit as shown in the adjoining diagram. When the frequency of the oscillation is increased, then at the resonant frequency, the voltmeter reading is zero in the case of<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/1676fe19-0483-4851-ac28-556e31160568-image.png" style="width: 280px; height: 201px;"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">voltmeter <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>1</mn></msub></math> only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">voltmeter <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>2</mn></msub></math> only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">voltmeter <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>3</mn></msub></math> only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All the three voltmeters</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">voltmeter <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>2</mn></msub></math> only</span> </div>

answer:
<div class="solution"><p>At resonance, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>L</mi></msub><mo>=</mo><msub><mi>X</mi><mi>C</mi></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mi>L</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mi>C</mi></mrow></mfrac></math></p><p>Voltage across the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>-</mo><mi>C</mi></math> combination,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><mi>i</mi><mfenced><mrow><msub><mi>X</mi><mi>L</mi></msub><mo>-</mo><msub><mi>X</mi><mi>C</mi></msub></mrow></mfenced><mo>=</mo><mn>0</mn></math></p></div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch1_P1.db
title:
a-rifle-man-who-together-with-his-rifle-has-a-mass-of-100-kg-stands-on-a-smooth-surface-and-fires-10-shots-horizontally-each-bullet-has-a-mass-10-g-an

slug:
a-rifle-man-who-together-with-his-rifle-has-a-mass-of-100-kg-stands-on-a-smooth-surface-and-fires-10-shots-horizontally-each-bullet-has-a-mass-10-g-an-57089

question:
<div class="question">A rifle man, who together with his rifle has a mass of $100 \mathrm{~kg}$, stands on a smooth surface and fires 10 shots horizontally. Each bullet has a mass $10 \mathrm{~g}$ and a muzzle velocity of $800 \mathrm{~ms}^{-1}$. The velocity which the rifle man attains after firing 10 shots is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 \mathrm{~ms}^{-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.8 \mathrm{~ms}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.08 \mathrm{~ms}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-0.8 \mathrm{~ms}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.8 \mathrm{~ms}^{-1}$</span> </div>

answer:
<div class="solution">According to law of conservation of momentum,
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>100 \mathrm{v}=-\frac{10}{1000} \times 10 \times 800 \\
<br/> <br/>\text { i.e., } \mathrm{v}=0.8 \mathrm{~ms}^{-1}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
title:
a-round-disc-of-moment-of-inertia-i-2-about-its-axis-perpendicular-to-its-plane-and-passing-through-its-centre-is-placed-over-another-disc-of-moment-o

slug:
a-round-disc-of-moment-of-inertia-i-2-about-its-axis-perpendicular-to-its-plane-and-passing-through-its-centre-is-placed-over-another-disc-of-moment-o-55194

question:
<div class="question">A round disc of moment of inertia $\mathrm{I}_{2}$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia $\mathrm{I}_{1}$ rotating with an angular velocity $\omega$ about the same axis. The final angular velocity of the combination of discs is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \omega}{\mathrm{I}_{1}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{I}_{2} \omega}{\mathrm{I}_{1}+\mathrm{I}_{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\omega$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{I}_{1} \omega}{\mathrm{I}_{1}+\mathrm{I}_{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{I}_{1} \omega}{\mathrm{I}_{1}+\mathrm{I}_{2}}$</span> </div>

answer:
<div class="solution">Total angular momentum of the system initially \(\mathrm{L}_{\mathrm{i}}=\mathrm{I}_{1} \mathrm{w}+\mathrm{I}_{2}(0)=\mathrm{I}_{1} \mathrm{w}\)
<br/> <br/>Total angular momentum of the system finally \(L_{f}=\left(I_{1}+I_{2}\right) w_{2}\)
<br/> <br/>According to conservation of angular momentum i.e. \(\mathrm{L}_{\mathrm{i}}=\mathrm{L}_{\mathrm{f}}\)
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\Rightarrow \mathrm{I}_{1} \omega=\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \omega_{2} \\
<br/> <br/>&amp;\Rightarrow \omega_{2}=\frac{\mathrm{I}_{1} \omega}{\mathrm{I}_{1}+\mathrm{I}_{2}}
<br/> <br/>\end{aligned}\)</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch1_P1.db
title:
a-rubber-cord-has-a-crosssectional-area-10-6-m-2-and-total-unstretched-length-0-1-m-it-is-stretched-to-0-125-m-and-then-released-to-project-a-particle

slug:
a-rubber-cord-has-a-crosssectional-area-10-6-m-2-and-total-unstretched-length-0-1-m-it-is-stretched-to-0-125-m-and-then-released-to-project-a-particle-18735

question:
<div class="question">A rubber cord has a cross-sectional area <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow><mo>-</mo><mn>6</mn></mrow></msup><mo> </mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math> and total unstretched length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math>. It is stretched to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>125</mn><mo> </mo><mi mathvariant="normal">m</mi></math> and then released to project a particle of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>0</mn><mo> </mo><mi mathvariant="normal">g</mi></math>. The velocity of projection is [Given, Young's modulus of rubber, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo> </mo><mi mathvariant="normal">N</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>]</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><p>Force constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math> of rubber is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo>=</mo><mfrac><mrow><mi>Y</mi><mi>A</mi></mrow><mi>l</mi></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>6</mn></mrow></msup></mrow><mrow><mn>0</mn><mo>.</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi mathvariant="normal">N</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Now, from conservation of energy,</p><p>elastic potential energy of cord <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> kinetic energy of particle</p><p>i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>K</mi><msup><mfenced><mrow><mo>∆</mo><mi>l</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>v</mi><mo>=</mo><msqrt><mfrac><mi>K</mi><mi>m</mi></mfrac></msqrt><mo>·</mo><mo>∆</mo><mi>l</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup></mrow><mrow><mn>5</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfrac></msqrt><mfenced><mrow><mn>0</mn><mo>.</mo><mn>125</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mn>10</mn><mn>3</mn></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>025</mn><mo>=</mo><mn>25</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p></div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P1.db
title:
a-running-track-of-440-ft-is-to-be-laid-out-enclosing-a-football-field-the-shape-of-which-is-a-rectangle-with-a-semicircle-at-each-end-if-the-area-of-

slug:
a-running-track-of-440-ft-is-to-be-laid-out-enclosing-a-football-field-the-shape-of-which-is-a-rectangle-with-a-semicircle-at-each-end-if-the-area-of-95051

question:
<div class="question">A running track of $440 \mathrm{ft}$ is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$70 \mathrm{ft}$ and $110 \mathrm{ft}$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$80 \mathrm{ft}$ and $120 \mathrm{ft}$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$35 \mathrm{ft}$ and $110 \mathrm{ft}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$35 \mathrm{ft}$ and $120 \mathrm{ft}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$35 \mathrm{ft}$ and $110 \mathrm{ft}$
<br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/aibsGJb8SAnd0AlzAIOjNYoNfQTsHZOd1upqzr5sx14.original.fullsize.png"/><br/> <br/>Perimeter $=440 \mathrm{ft}$
<br/>$\Rightarrow \quad 2 x+\pi r+\pi r=440$
<br/>$2 x+2 \pi r=440$
<br/>$$
<br/>\begin{aligned}
<br/>A &amp; =\text { Area of the rectangular portion }=x 2 r \\
<br/>A &amp; =x \frac{(440-2 x)}{\pi}=\frac{1}{\pi}\left(440 x-2 x^2\right)
<br/>\end{aligned}
<br/>$$
<br/>Let $\frac{d A}{d x}=\frac{1}{\pi}(440-4 x)=0$
<br/>$$
<br/>\begin{aligned}
<br/>&amp; \Rightarrow x=110 \text { for which } \frac{d^2 A}{d x^2} &lt; 0 \\
<br/>&amp; \Rightarrow A \text { is maximum when } x=110 \\
<br/>&amp; \Rightarrow 2 r=\frac{440-2 x}{\pi}=\frac{440-220}{22 / 7}=70 \\
<br/>&amp; \therefore \quad r=35 \mathrm{ft} \text { and } x=110 \mathrm{ft}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch1_P1.db
title:
a-sample-of-gas-expands-from-volume-v-1-to-v-2-the-amount-of-work-done-by-the-gas-is-greatest-when-the-expansion-is

slug:
a-sample-of-gas-expands-from-volume-v-1-to-v-2-the-amount-of-work-done-by-the-gas-is-greatest-when-the-expansion-is-88901

question:
<div class="question">A sample of gas expands from volume $\mathrm{V}_{1}$ to $\mathrm{V}_{2}$. The amount of work done by the gas is greatest when the expansion is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">isothermal</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">isobaric</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">adiabatic</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">equal in all cases</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">isothermal</span> </div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch1_P1.db
title:
a-sample-of-ideal-monoatomic-gas-is-taken-round-the-cycle-abca-as-shown-in-the-figure-the-work-done-during-the-cycle-is

slug:
a-sample-of-ideal-monoatomic-gas-is-taken-round-the-cycle-abca-as-shown-in-the-figure-the-work-done-during-the-cycle-is-71357

question:
<div class="question">A sample of ideal monoatomic gas is taken round the cycle ABCA as shown in the figure. The work done during the cycle is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Dn4mh3Fq-NJiyG35Ir_DMNaX9dtBaVHvH36wSjlDcKo.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 \mathrm{pV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$9 \mathrm{pV}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$6 \mathrm{pV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \mathrm{pV}$</span> </div>

answer:
<div class="solution">$\Delta \mathrm{W}=$ area under the $\mathrm{p}-\mathrm{V}$ curve
<br/>$=\frac{1}{2} \times 3 \mathrm{p} \times 2 \mathrm{~V}=3 \mathrm{pV}$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch1_P1.db
title:
a-semicircular-disc-of-mass-m-and-radius-r-is-free-to-rotate-about-its-diameter-the-moment-of-inertia-of-semicircular-disc-about-a-line-perpendicular-

slug:
a-semicircular-disc-of-mass-m-and-radius-r-is-free-to-rotate-about-its-diameter-the-moment-of-inertia-of-semicircular-disc-about-a-line-perpendicular-22521

question:
<div class="question">A semicircular disc of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is free to rotate about its diameter. The moment of inertia of semicircular disc about a line perpendicular to its plane through centre is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>The moment of inertia of disc of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> about its axis is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mfrac><mrow><msup><mi>M</mi><mo>'</mo></msup><msup><mi>R</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>M</mi><mo>'</mo></msup><mo>=</mo></math> mass of complete disc <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mi>M</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>1</mn></msub></math> be the moment of inertia of semicircular disc about its. axis, then disc may be assumed as combination of two semi-circular parts,</p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><msub><mi>I</mi><mn>1</mn></msub><mo>+</mo><msub><mi>I</mi><mn>1</mn></msub><mo>⇒</mo><msub><mi>I</mi><mn>1</mn></msub><mo>=</mo><mfrac><mi>I</mi><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>M</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac></math></p></div>


tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch1_P1.db
title:
a-semicircular-wire-of-radius-r-rotates-in-uniform-magnetic-field-b-about-its-diameter-with-angular-velocity-if-the-total-resistance-of-the-circuit-is

slug:
a-semicircular-wire-of-radius-r-rotates-in-uniform-magnetic-field-b-about-its-diameter-with-angular-velocity-if-the-total-resistance-of-the-circuit-is-73272

question:
<div class="question">A semicircular wire of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> rotates in uniform magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> about its diameter with angular velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>. If the total resistance of the circuit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>, then the mean power generated per period of rotation is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>π</mi><mi>B</mi><msup><mi>r</mi><mn>2</mn></msup><mi>ω</mi></mrow></mfenced><mn>2</mn></msup><mrow><mn>8</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>π</mi><mi>B</mi><mi>r</mi><msup><mi>ω</mi><mn>2</mn></msup></mrow><mrow><mn>8</mn><mi>R</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>π</mi><mi>B</mi><mi>r</mi><mi>ω</mi></mrow></mfenced><mn>2</mn></msup><mrow><mn>8</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>π</mi><mi>B</mi><msup><mi>r</mi><mn>2</mn></msup><mi>ω</mi></mrow></mfenced><mn>2</mn></msup><mrow><mn>8</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p>The flux linked with coil (made of wire of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>) of area <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mi>B</mi><mi>A</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mi>B</mi><mo> </mo><mfenced><mfrac><mrow><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac></mfenced><mo> </mo><mi>cos</mi><mo> </mo><mi>ω</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mfrac><mrow><mi>π</mi><mi>B</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo><mi>cos</mi><mo> </mo><mi>ω</mi><mi>t</mi></math>   [<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>A</mi><mo>=</mo><mfrac><mrow><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mi>ω</mi><mi>t</mi></math>]</p><p>By Faraday's law of electromagnetic induction, induced emf, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>d</mi><mi>ϕ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math></p><p>Hence, after differentiating,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mo>-</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mfenced><mrow><mfrac><mrow><mi>π</mi><mi>B</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo><mi>cos</mi><mo> </mo><mi>ω</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>π</mi><mi>B</mi><msup><mi>r</mi><mn>2</mn></msup><mi>ω</mi></mrow><mn>2</mn></mfrac><mo> </mo><mi>sin</mi><mo> </mo><mi>ω</mi><mi>t</mi></math>  (where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo></math> resistance in the circuit)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Power <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><msup><mi>e</mi><mn>2</mn></msup><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><msup><mi>π</mi><mn>2</mn></msup><msup><mi>B</mi><mn>2</mn></msup><msup><mi>r</mi><mn>4</mn></msup><msup><mi>ω</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mi>ω</mi><mi>t</mi></mrow><mrow><mn>4</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Mean power <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>P</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><msup><mi>π</mi><mn>2</mn></msup><msup><mi>B</mi><mn>2</mn></msup><msup><mi>r</mi><mn>4</mn></msup><msup><mi>ω</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac><mo>·</mo><mover><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi>ω</mi><mi>t</mi></mrow><mo>¯</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>π</mi><mn>2</mn></msup><msup><mi>B</mi><mn>2</mn></msup><msup><mi>r</mi><mn>4</mn></msup><msup><mi>ω</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac><mo>·</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mover><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi>ω</mi><mi>t</mi></mrow><mo>¯</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msup><mfenced><mrow><mi>π</mi><mi>B</mi><msup><mi>r</mi><mn>2</mn></msup><mi>ω</mi></mrow></mfenced><mn>2</mn></msup><mrow><mn>8</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac></math></p></div>


title: a-sequence-of-how-many-nucleotides-in-messenger-rna-makes-a-codon-for-an-amino-acid
slug: a-sequence-of-how-many-nucleotides-in-messenger-rna-makes-a-codon-for-an-amino-acid-26634
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Three</span> </div>
source_db: MarksBatch1_P1.db
question:
<div class="question">A sequence of how many nucleotides in messenger RNA makes a codon for an amino acid?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Three</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Four</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">One</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Two</span> </li> </ul>

answer:
<div class="solution">The sequence of bases in mRNA are read in a serial order in groups of three at a time. Each triplet of nucleotides (having a specific sequence of bases) in known as codon. Each codon specifies one amino acid. Further since, there are four bases. therefore, $4^{3}=64$ triplets or codons are possible.</div>


title: a-set-a-has-3-elements-and-another-set-b-has-6-elements-then
slug: a-set-a-has-3-elements-and-another-set-b-has-6-elements-then-88863
question: <div class="question">A set $A$ has 3 elements and another set B has 6 elements. Then</div>
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch1_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \leq \mathrm{n}(\mathrm{A} \cup \mathrm{B}) \leq 6$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \leq \mathrm{n}(\mathrm{A} \cup \mathrm{B}) \leq 9$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$6 \leq \mathrm{n}(\mathrm{A} \cup \mathrm{B}) \leq 9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0 \leq n(A \cup B) \leq 9$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6 \leq \mathrm{n}(\mathrm{A} \cup \mathrm{B}) \leq 9$</span> </div>

answer:
<div class="solution">We have<br/>$\min n(A \cup B)=\max \{n(A), n(B)\}=\max \{3,6\}=6$<br/>$\operatorname{Max} n(A \cup B)=n(A)+n(B)=9$<br/>$\therefore 6 \leq n(A \cup B) \leq 9$</div>


tags: ['Mathematics', 'Linear Programming', 'BITSAT', 'BITSAT 2013']
answer: <div class="solution">$x, y \geq 0$ and $4 x+3 y \leq 24$.</div>
source_db: MarksBatch1_P1.db
title:
a-shopkeeper-wants-to-purchase-two-articles-a-and-b-of-cost-price-4-and-3-respectively-he-thought-that-he-mayearn-30-paise-byselling-article-a-and-10-

slug:
a-shopkeeper-wants-to-purchase-two-articles-a-and-b-of-cost-price-4-and-3-respectively-he-thought-that-he-mayearn-30-paise-byselling-article-a-and-10-27979

question:
<div class="question">A shopkeeper wants to purchase two articles A and $\mathrm{B}$ of cost price 4 and 3 respectively. He thought that he mayearn 30 paise byselling article A and 10 paise by selling article $B$. He has not to purchase total articles worth more than ' 24 . If he purchases the number of articles of $\mathrm{A}$ and $\mathrm{B}, \mathrm{x}$ and y respectively, then linear constraints are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x \geq 0, y \geq 0,4 x+3 y \leq 24$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x \geq 0, y \geq 0,30 x+10 y \leq 24$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x \geq 0, y \geq 0,4 x+3 y \geq 24$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x \geq 0, y \geq 0,30 x+40 y \geq 24$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x \geq 0, y \geq 0,4 x+3 y \leq 24$</span> </div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch1_P1.db
title:
a-simple-pendulum-has-time-period-t-its-time-period-in-a-lift-which-is-moving-upwards-with-acceleration-3-ms-2-is

slug:
a-simple-pendulum-has-time-period-t-its-time-period-in-a-lift-which-is-moving-upwards-with-acceleration-3-ms-2-is-38961

question:
<div class="question">A simple pendulum has time period 't'. Its time period in a lift which is moving upwards with acceleration $3 \mathrm{~ms}^{-2}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$t \sqrt{\frac{9.8}{12.8}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$t \sqrt{\frac{12.8}{9.8}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{t} \sqrt{\frac{9.8}{6.8}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{t} \sqrt{\frac{6.8}{9.8}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$t \sqrt{\frac{9.8}{12.8}}$</span> </div>

answer:
<div class="solution">$\mathrm{t} \propto \frac{1}{\sqrt{9.8}}, \mathrm{t}^{\prime} \propto \frac{1}{\sqrt{12.8}}$<br/>$\left(\because \mathrm{g}^{\prime}=9.8+3=12.8\right)$<br/>$\therefore \frac{\mathrm{t}^{\prime}}{\mathrm{t}}=\sqrt{\frac{9.8}{12.8}} \Rightarrow \mathrm{t}^{\prime}=\sqrt{\frac{9.8}{12.8}} \mathrm{t}$</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch1_P1.db
title:
a-single-slit-of-a-width-a-is-illuminated-by-a-monochromatic-light-of-wavelength-600-nm-the-value-of-a-for-which-first-minimum-appears-at-3-0-on-the-s

slug:
a-single-slit-of-a-width-a-is-illuminated-by-a-monochromatic-light-of-wavelength-600-nm-the-value-of-a-for-which-first-minimum-appears-at-3-0-on-the-s-34851

question:
<div class="question">A single slit of a width $a$ is illuminated by a monochromatic light of wavelength $600 \mathrm{~nm}$. The value of ' $a$ ' for which first minimum appears at $\theta$ $=30^{\circ}$ on the screen will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.6 \mu \mathrm{m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.2 \mu \mathrm{m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.8 \mu \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \mu \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.2 \mu \mathrm{m}$</span> </div>

answer:
<div class="solution">Wavelength of monochromatic light,<br/>$\lambda=600 \times 10^{-9} \mathrm{~m}$<br/>As fir first minima a $\sin \theta=\lambda$<br/>$\begin{aligned}<br/>&amp; \Rightarrow \mathrm{a} \sin 30^{\circ}=600 \times 10^{-9} \\<br/>&amp; \Rightarrow \mathrm{a}=1200 \times 10^{-9} \mathrm{~m}=1.2 \mu \mathrm{m}<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P1.db
title:
a-six-faced-die-is-a-biased-one-it-is-thrice-more-likely-to-show-an-odd-numbers-than-show-an-even-number-it-is-thrown-twice-the-probability-that-the-s

slug:
a-six-faced-die-is-a-biased-one-it-is-thrice-more-likely-to-show-an-odd-numbers-than-show-an-even-number-it-is-thrown-twice-the-probability-that-the-s-51055

question:
<div class="question">A six faced die is a biased one. It is thrice more likely to show an odd numbers than show an even number. It is thrown twice. The probability that the sum of the numbers in two throws is even, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{5}{9}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{5}{8}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{1}{2}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{5}{8}$
<br/></span> </div>

answer:
<div class="solution">Let $P$ be the probability of getting an even number. Then by hypothesis the probability of getting an odd number is $3 P$. Since the events of getting an even number and an odd number are mutually exclusive and exhaustive.
<br/>$$
<br/>\therefore \quad P+3 P=1 \Rightarrow P=1 / 4
<br/>$$
<br/>Thus, the probability of getting an odd number in a single throw is $\frac{3}{4}$ and that of an even number is $\frac{1}{4}$.
<br/>If the die is thrown twice, then the sum of the numbers in two throws is even if both the numbers are even or both are odd.
<br/>$$
<br/>\therefore \text { Required probability }=\frac{3}{4} \times \frac{3}{4}+\frac{1}{4} \times \frac{1}{4}=\frac{10}{16}=\frac{5}{8}
<br/>$$</div>


tags: ['Physics', 'Work Power Energy', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch1_P1.db
title:
a-small-block-of-mass-m-is-kept-on-a-rough-inclined-surface-of-inclination-fixed-in-a-elevator-the-elevator-goes-up-with-a-uniform-velocity-v-and-the-

slug:
a-small-block-of-mass-m-is-kept-on-a-rough-inclined-surface-of-inclination-fixed-in-a-elevator-the-elevator-goes-up-with-a-uniform-velocity-v-and-the-81921

question:
<div class="question">A small block of mass $\mathrm{m}$ is kept on a rough inclined surface of inclination $\theta$ fixed in a elevator. The elevator goes up with a uniform velocity $\mathrm{v}$ and the block does not slide on the wedge. The work done by the force of friction on the block in time t will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{mgvt} \cos ^{2} \theta$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{mgvt} \sin ^{2} \theta$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{mgvt} \sin 2 \theta$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{mgvt} \sin ^{2} \theta$</span> </div>

answer:
<div class="solution">\(\mathrm{W}_{\mathrm{f}}=\mathrm{f} \sin \theta \mathrm{S}\)
<br/> <br/>\(=(\mathrm{mg} \sin \theta)(\sin \theta)(\mathrm{vt})\)
<br/> <br/>\(=\mathrm{mg} \sin ^{2} \theta \mathrm{vt}\)</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
title:
a-solenoid-is-2-m-long-and-3-cm-in-diameter-it-has-5-layers-of-winding-of-1000-turns-each-and-carries-a-current-of-5-a-what-is-the-magnetic-field-at-i

slug:
a-solenoid-is-2-m-long-and-3-cm-in-diameter-it-has-5-layers-of-winding-of-1000-turns-each-and-carries-a-current-of-5-a-what-is-the-magnetic-field-at-i-67174

question:
<div class="question">A solenoid is $2 \mathrm{~m}$ long and $3 \mathrm{~cm}$ in diameter. It has 5 layers of winding of 1000 turns each and carries a current of $5 \mathrm{~A}$. What is the magnetic field at its centre ? Use the standard value of $\mu_{0}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.57 \times 10^{-4} \mathrm{~T}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.57 \times 10^{-3} \mathrm{~T}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1.57 \times 10^{-2} \mathrm{~T}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3.57 \times 10^{-6} \mathrm{~T}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.57 \times 10^{-2} \mathrm{~T}$</span> </div>

answer:
<div class="solution">Length of solenoid, $l=2 \mathrm{~m}$
<br/> <br/>Total number of turns, $\mathrm{N}=5 \times 1000=5000$
<br/> <br/>Number of turns per unit length,
<br/> <br/>$\mathrm{n}=\frac{\mathrm{N}}{l}=\frac{5000}{2}=2500 \mathrm{~m}^{-1}$
<br/> <br/>Current, I = 5A
<br/> <br/>Now, B $=\mu_{0} \mathrm{nI}=4 \pi \times 10^{-7} \times 2500 \times 5 \mathrm{~T}$
<br/> <br/>$=1.57 \times 10^{-2} \mathrm{~T}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch1_P1.db
title:
a-solenoid-of-1200-turns-is-wound-uniformly-in-a-single-layer-on-a-glass-tube-2-m-long-and-02-m-in-diameter-the-magnetic-intensity-at-the-center-of-th

slug:
a-solenoid-of-1200-turns-is-wound-uniformly-in-a-single-layer-on-a-glass-tube-2-m-long-and-02-m-in-diameter-the-magnetic-intensity-at-the-center-of-th-75560

question:
<div class="question">A solenoid of 1200 turns is wound uniformly in a single layer on a glass tube $2 \mathrm{~m}$ long and $0.2 \mathrm{~m}$ in diameter. The magnetic intensity at the center of the solenoid when a current of 2 A flows through it is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.4 \times 10^3 \mathrm{Am}^{-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.2 \times 10^3 \mathrm{Am}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 \mathrm{Am}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.4 \times 10^{-3} \mathrm{~A} \mathrm{~m}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.2 \times 10^3 \mathrm{Am}^{-1}$</span> </div>

answer:
<div class="solution">Given that number of turns, $n=1200$<br/>Current, $\mathrm{I}=2 \mathrm{~A}$<br/>Magnetic field at centre inside the solenoid is given by,<br/>$\mathrm{B}=\mu_0 \mathrm{nI}$<br/>So magnetic intensity at centre of the solenoid,<br/>$\begin{aligned}<br/>&amp; \mathrm{H}=\frac{\mathrm{B}}{\mu_0}=\mathrm{nI}=\left(\frac{1200}{2}\right)(2) \quad\left(\because \mathrm{B}=\mu_0 \mathrm{H}\right) \\<br/>&amp; \mathrm{H}=1.2 \times 10^3 \mathrm{Am}^{-1}<br/>\end{aligned}$</div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch1_P1.db
title:
a-solid-body-of-constant-heat-capacity-1-j-c-is-being-heated-by-keeping-it-in-contact-with-reservoirs-in-two-ways-i-sequentially-keeping-in-contact-wi

slug:
a-solid-body-of-constant-heat-capacity-1-j-c-is-being-heated-by-keeping-it-in-contact-with-reservoirs-in-two-ways-i-sequentially-keeping-in-contact-wi-82889

question:
<div class="question">A solid body of constant heat capacity $1 \mathrm{~J} /{ }^{\circ} \mathrm{C}$ is being heated by keeping it in contact with reservoirs in two ways:
<br/>(i) Sequentially keeping in contact with 2 reservoirs such that each reservoir supplies same amount of heat.
<br/>(ii) Sequentially keeping in contact with 8 reservoirs such that each reservoir supplies same amount of heat. 
<br/>In both the cases body is brought from initial temperature $100^{\circ} \mathrm{C}$ to final temperature $200^{\circ} \mathrm{C}$. Entropy change of the body in the two cases respectively is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\ln 2,2 \ln 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \ln 2,8 \ln 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\ln 2,4 \ln 2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\ln 2, \ln 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\ln 2, \ln 2$</span> </div>

answer:
<div class="solution">The entropy change of the body in the two cases is same as entropy is a state function.</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch1_P1.db
title:
a-solid-cylinder-and-a-hollow-cylinder-both-of-the-same-mass-and-same-external-diameter-are-released-from-the-same-height-at-the-same-time-on-an-incli

slug:
a-solid-cylinder-and-a-hollow-cylinder-both-of-the-same-mass-and-same-external-diameter-are-released-from-the-same-height-at-the-same-time-on-an-incli-77429

question:
<div class="question">A solid cylinder and a hollow cylinder both of the same mass and same external diameter are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Solid cylinder</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Both together</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">One with higher density</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Hollow cylinder</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Solid cylinder</span> </div>

answer:
<div class="solution">Solid cylinder reaches the bottom first because for solid cylinder $\frac{\mathrm{K}^{2}}{\mathrm{R}^{2}}=\frac{1}{2}$ and for hollow cylinder, $\frac{\mathrm{K}^{2}}{\mathrm{R}^{2}}=1$.
<br/> <br/>Acceleration down the inclined plane $\propto \frac{1}{1+K^{2} / R^{2}} .$ Solid cylinder has greater acceleration. It reaches the bottom first.</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P1.db
title:
a-solid-sphere-of-80-kg-and-radius-15-m-moving-in-a-space-becomes-a-circular-disc-of-radius-20-m-in-1-h-the-rate-of-change-of-moment-of-inertia-in-thi

slug:
a-solid-sphere-of-80-kg-and-radius-15-m-moving-in-a-space-becomes-a-circular-disc-of-radius-20-m-in-1-h-the-rate-of-change-of-moment-of-inertia-in-thi-76786

question:
<div class="question">A solid sphere of $80 \mathrm{~kg}$ and radius $15 \mathrm{~m}$ moving in a space becomes a circular disc of radius $20 \mathrm{~m}$ in $1 \mathrm{~h}$. The rate of change of moment of Inertia in this process is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{30}{9} \mathrm{~kg}^2 \mathrm{~m}^2 \mathrm{~s}^{-1}$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{25}{9} \mathrm{~kg}^{-\mathrm{m}^2 \mathrm{~s}^{-1}}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{10}{9} \mathrm{~kg}-\mathrm{m}^2 \mathrm{~s}^{-1}$
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>$\frac{22}{9} \mathrm{~kg}^{-\mathrm{m}^2 \mathrm{~s}^{-1}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{22}{9} \mathrm{~kg}^{-\mathrm{m}^2 \mathrm{~s}^{-1}}$</span> </div>

answer:
<div class="solution">Given, mass of solid sphere $=80 \mathrm{~kg}$ radius of solid sphere, $R_s=15 \mathrm{~m}$
<br/>radius of circular disc, $R_c=20 \mathrm{~m}$ and time $=1$ hour $=60$ minute $=60 \times 60 \mathrm{sec}$
<br/>$\therefore$ Moment of inertia of solid sphere, $I_s=\frac{2}{5} M R^2$
<br/>$$
<br/>\begin{aligned}
<br/>&amp; =\frac{2}{5} \times 80 \times(15)^2 \\
<br/>&amp; =7200 \mathrm{~kg}-\mathrm{m}^2
<br/>\end{aligned}
<br/>$$
<br/>Similarly,
<br/>$$
<br/>\text { moment of inertia of the disc, } \begin{aligned}
<br/>I_c &amp; =\frac{1}{2} M R_c^2 \\
<br/>&amp; =\frac{1}{2} \times 80 \times(20)^2 \\
<br/>&amp; =16000 \mathrm{~kg}-\mathrm{m}^2
<br/>\end{aligned}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>\text { Rate of change of moment of Inertia } &amp; =\frac{I_c-I_s}{t} \\
<br/>&amp; =\frac{16000-7200}{60 \times 60} \\
<br/>&amp; =\frac{22}{9} \mathrm{~kg}-\mathrm{m}^2 \mathrm{~s}^{-1}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch1_P1.db
title:
a-solid-sphere-of-radius-r-has-a-charge-q-distributed-in-its-volume-with-a-charge-density-k-r-a-where-k-and-a-are-constants-and-r-is-the-distance-from-1

slug:
a-solid-sphere-of-radius-r-has-a-charge-q-distributed-in-its-volume-with-a-charge-density-k-r-a-where-k-and-a-are-constants-and-r-is-the-distance-from-1-90008

question:
<div class="question">A solid sphere of radius $R$ has a charge $Q$ distributed in its volume with a charge density $\rho$ $=k r^{a}$, where $k$ and $a$ are constants and $r$ is the distance from its centre. If the electric field at $r=$ $\frac{R}{2}$ is $\frac{1}{8}$ times that at $r=R,$ the value of $a$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7</span> </li> </ul>

answer:
<div class="solution">Using Gauss's law, we have
<br/> <br/>$$
<br/> <br/>\oint \vec{E} \cdot d \vec{A}=\frac{1}{\epsilon_{0}} \int(\rho d v)=\frac{1}{\epsilon_{0}} \int_{0}^{R} k r^{a} \times 4 \pi r^{2} d r
<br/> <br/>$$
<br/> <br/>or $E \times 4 \pi R^{2}=\left(\frac{4 \pi k}{\epsilon_{0}}\right) \frac{R^{(a+3)}}{(a+3)}$
<br/> <br/>$$
<br/> <br/>\therefore \quad E_{1}=\frac{k R^{(a+1)}}{\epsilon_{0}(a+3)}
<br/> <br/>$$
<br/> <br/>For $r=\frac{R}{2} \cdot E_{2}=\frac{k\left(\frac{R}{2}\right)^{a+1}}{\epsilon_{0}(a+3)}$
<br/> <br/>Given, $E_{2}=\frac{E_{1}}{8}$
<br/> <br/>or $\frac{k\left(\frac{R}{2}\right)^{a+1}}{\epsilon_{0}(a+3)}=\frac{1}{8} \frac{k R^{(a+1)}}{\epsilon_{0}(a+3)}$
<br/> <br/>$$
<br/> <br/>\begin{array}{ll}
<br/> <br/>\therefore &amp; \quad \frac{1}{a+1}=\frac{1}{8} \\
<br/> <br/>\text { or } &amp; a=2 .
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch1_P1.db
title:
a-solution-of-nh-4-cl-and-nh-3-has-ph-80-which-of-the-following-hydroxides-may-be-precipitated-when-this-solution-is-mixed-with-equal-volume-of-02-m-o

slug:
a-solution-of-nh-4-cl-and-nh-3-has-ph-80-which-of-the-following-hydroxides-may-be-precipitated-when-this-solution-is-mixed-with-equal-volume-of-02-m-o-74365

question:
<div class="question">A solution of $\mathrm{NH}_{4} \mathrm{Cl}$ and $\mathrm{NH}_{3}$ has $\mathrm{pH}=8.0 .$ Which of the following hydroxides may be precipitated when this solution is mixed with equal volume of 0.2 M of metal ion.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Ba}(\mathrm{OH})_{2}\left(\mathrm{~K}_{\mathrm{sp}}=1.1 \times 10^{-4}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Mg}(\mathrm{OH})_{2}\left(\mathrm{~K}_{\mathrm{sp}}=3.5 \times 10^{-4}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{Fe}(\mathrm{OH})_{2}\left(\mathrm{~K}_{\mathrm{sp}}=8.1 \times 10^{-16}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Ca}(\mathrm{OH})_{2}\left(\mathrm{~K}_{\mathrm{sp}}=2.1 \times 10^{-5}\right)$ ).</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Fe}(\mathrm{OH})_{2}\left(\mathrm{~K}_{\mathrm{sp}}=8.1 \times 10^{-16}\right)$</span> </div>

answer:
<div class="solution">$\mathrm{pH}=8, \mathrm{pOH}=6 ;\left[\mathrm{OH}^{-}\right]=10^{-6} \mathrm{M}$
<br/> <br/>Ionic product of $\mathrm{Fe}(\mathrm{OH})_{2}=0.2 \times\left(1 \times 10^{-6}\right)^{2}$
<br/> <br/>$=2 \times 10^{-13}&gt;\mathrm{K}_{\mathrm{sp}}\left(=8.1 \times 10^{-16}\right)$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch1_P1.db
title:
a-sound-souorce-is-moving-towards-stationary-listener-with-10-1-th-of-the-speed-of-sound-the-ratio-of-apparent-to-read-frequency-is

slug:
a-sound-souorce-is-moving-towards-stationary-listener-with-10-1-th-of-the-speed-of-sound-the-ratio-of-apparent-to-read-frequency-is-73348

question:
<div class="question">A sound souorce is moving towards stationary listener with $\frac{1}{10}$ th of the speed of sound. The ratio of apparent to read frequency is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\frac{9}{10}\right)^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\frac{10}{9}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{11}{10}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{11}{10}\right)^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{10}{9}\right)$</span> </div>

answer:
<div class="solution">$v_{\mathrm{s}}=\frac{\mathrm{v}}{10} \quad \mathrm{n}^{\prime}=\mathrm{n} \frac{\mathrm{v}}{\mathrm{v}-\mathrm{v}_{\mathrm{s}}}$
<br/>$\frac{\mathrm{n}^{\prime}}{\mathrm{n}}=\frac{\mathrm{v}}{\left(\mathrm{v}-\frac{\mathrm{v}}{10}\right)}=\frac{10}{9}$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2014']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P1.db
title:
a-sound-source-emitting-sound-of-constant-frequency-moves-with-a-constant-speed-and-crosses-a-stationary-observer-the-frequency-n-of-sound-heard-by-th

slug:
a-sound-source-emitting-sound-of-constant-frequency-moves-with-a-constant-speed-and-crosses-a-stationary-observer-the-frequency-n-of-sound-heard-by-th-88562

question:
<div class="question">A sound source, emitting sound of constant frequency, moves with a constant speed and crosses a stationary observer. The frequency (n) of sound heard by the observer is plotted against time (t). Which of the following graphs represents the correct variation?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/dxgjh0YqWHXzF8iEaFK08UYDQAykQN_t-h42ClX36r0.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/AG9jrl0KXcABKQf8Tqq6NpI8ldsZMHF4_9eLyamyihE.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Klnuor_6ZFyxfO6yDkahYUcNnBjcNnU9ePm6J1Uvsd0.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/MSPthFQUS_ZP9AARTiTCNWiYDHZRXNb7cfbhaG-lR9I.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/MSPthFQUS_ZP9AARTiTCNWiYDHZRXNb7cfbhaG-lR9I.original.fullsize.png"/></span> </div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2010']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch1_P1.db
title:
a-source-of-sound-produces-waves-of-wavelength-60-cm-when-it-is-stationary-if-the-speed-of-sound-in-air-is-320-m-s-1-and-source-moves-with-speed-20-m-

slug:
a-source-of-sound-produces-waves-of-wavelength-60-cm-when-it-is-stationary-if-the-speed-of-sound-in-air-is-320-m-s-1-and-source-moves-with-speed-20-m-83874

question:
<div class="question">A source of sound produces waves of wavelength $60 \mathrm{~cm}$ when it is stationary. If the speed of sound in air is $320 \mathrm{~m} \mathrm{~s}^{-1}$ and source moves with speed $20 \mathrm{~m} \mathrm{~s}^{-1},$ the wavelength of sound in the forward direction will be nearest to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$56 \mathrm{~cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$60 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$64 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$68 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$56 \mathrm{~cm}$</span> </div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch1_P1.db
title:
a-source-of-sound-s-emitting-waves-of-frequency-100-hz-and-an-observor-o-are-located-at-some-distance-from-each-other-the-source-is-moving-with-a-spee

slug:
a-source-of-sound-s-emitting-waves-of-frequency-100-hz-and-an-observor-o-are-located-at-some-distance-from-each-other-the-source-is-moving-with-a-spee-23715

question:
<div class="question">A source of sound S emitting waves of frequency $100 \mathrm{~Hz}$ and an observor $\mathrm{O}$ are located at some distance from each other. The source is moving with a speed of $19.4 \mathrm{~ms}^{-1}$ at an angle of $60^{\circ}$ with the source observer line as shown in the figure. The observor is at rest. The apparent frequency observed by the observer is (velocity of sound in air $330 \mathrm{~ms}^{-1}$ )
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/gzDojaZnw4cKWu4LevpU1GrsaIDMmmjIgy7sxqgc_SM.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$103 \mathrm{~Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$106 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$97 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$100 \mathrm{~Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$103 \mathrm{~Hz}$</span> </div>

answer:
<div class="solution">Here, original frequency of sound, $\mathrm{f}_{0}=100 \mathrm{~Hz}$ Speed of source $V_{s}=19.4 \cos 60^{\circ}=9.7$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/kl3r4C7gEpsUBy2rdOIXHWvaF48AGvWkG4pVu5UMWEE.original.fullsize.png"/><br/> <br/> <br/>From Doppler's formula
<br/> <br/>$\mathrm{f}^{\mathrm{l}}=\mathrm{f}_{0}\left(\frac{\mathrm{V}-\mathrm{V}_{0}}{\mathrm{~V}-\mathrm{V}_{\mathrm{s}}}\right)$
<br/> <br/>$\mathrm{f}^{\mathrm{l}}=100\left(\frac{\mathrm{V}-0}{\mathrm{~V}-(+9.7)}\right)$
<br/> <br/>$\mathrm{f}^{\mathrm{l}}=100 \frac{\mathrm{V}}{\mathrm{V}\left(1-\frac{9.7}{\mathrm{~V}}\right)}$
<br/> <br/>$\mathrm{f}^{1}=100\left(1+\frac{9.7}{330}\right)=103 \mathrm{~Hz}$
<br/> <br/>Apparent frequency $\mathrm{f}^{1}=103 \mathrm{~Hz}$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2010']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch1_P1.db
title:
a-spherical-ball-is-dropped-in-a-long-column-of-a-viscous-liquid-the-speed-v-of-the-ball-as-a-function-of-time-t-may-be-best-represented-by

slug:
a-spherical-ball-is-dropped-in-a-long-column-of-a-viscous-liquid-the-speed-v-of-the-ball-as-a-function-of-time-t-may-be-best-represented-by-94672

question:
<div class="question">A spherical ball is dropped in a long column of a viscous liquid. The speed (v) of the ball as a function of time (t) may be best represented by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/yA-XRpi5D97euZMz1k3L2ME1aRMPPsRi72AXDa4zprI.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/m8awBdQcBYhNqxGk_rcK2iqimxjFX6Nxvdbl_zNIF_U.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/bUC4hYOns3Bg0jo0q_NY7eqZr7CaR1YufQ5gOrnXpQY.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/LV3GllZxakI6groXRpGE5lyTCyRxxCyz5NZTqa9IzU8.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/m8awBdQcBYhNqxGk_rcK2iqimxjFX6Nxvdbl_zNIF_U.original.fullsize.png"/></span> </div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P1.db
title:
a-spherical-balloon-is-filled-with-4500-cubic-meters-of-helium-gas-if-a-leak-in-the-balloon-causes-the-gas-to-escape-at-the-rate-of-72-cubic-meters-pe-1

slug:
a-spherical-balloon-is-filled-with-4500-cubic-meters-of-helium-gas-if-a-leak-in-the-balloon-causes-the-gas-to-escape-at-the-rate-of-72-cubic-meters-pe-1-92081

question:
<div class="question">A spherical balloon is filled with $4500 ~\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 ~\pi$ cubic meters per minute then the rate (in meters per minute) at which the radius of the balloon decreases $49 \mathrm{~min}$ after the leakage began is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{9}{7}$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{7}{9}$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{2}{9}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$9$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{2}{9}$
<br/></span> </div>

answer:
<div class="solution">$V=\frac{4}{3} \pi r^3 \Rightarrow 4500 \pi=\frac{4 \pi r^3}{3}$
<br/>$\Rightarrow \quad r=15 \mathrm{~m}$
<br/>After $49 \min =(4500-49.72) \pi$
<br/>$=972 \pi \mathrm{m}^3$
<br/>$\begin{aligned} \Rightarrow &amp; &amp; 972 \pi &amp; =\frac{4}{3} \pi r^3 \\ \Rightarrow &amp; &amp; r^3 &amp; =3 \times 243=3 \times 3^5\end{aligned}$
<br/>$\begin{aligned} r &amp; =9 \\ \frac{d v}{d t} &amp; =4 \pi r^2\left(\frac{d r}{d t}\right) \\ 72 \pi &amp; =4 \pi \times 9 \times 9\left(\frac{d r}{d t}\right) \\ \frac{d r}{d t} &amp; =\left(\frac{2}{9}\right)\end{aligned}$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
title:
a-spherical-drop-of-radius-r-is-divided-into-eight-equal-droplets-if-surface-tension-is-t-then-the-work-done-in-this-process-is

slug:
a-spherical-drop-of-radius-r-is-divided-into-eight-equal-droplets-if-surface-tension-is-t-then-the-work-done-in-this-process-is-61091

question:
<div class="question">A spherical drop of radius $R$ is divided into eight equal droplets. If surface tension is $\mathrm{T}$, then the work done in this process is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \pi \mathrm{R}^{2} \mathrm{~T}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \pi R^{2} T$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4 \pi R^{2} T$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \pi \mathrm{RT}^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \pi R^{2} T$</span> </div>

answer:
<div class="solution">Volume of 8 small droplets $=$ Volume of 1 big drop $\Rightarrow \frac{4}{3} \pi r^{3} \times 8=\frac{4}{3} \pi \mathrm{R}^{3} \Rightarrow r=\frac{\mathrm{R}}{(8)^{1 / 3}}$
<br/> <br/>Work done $=($ Change in area $) \times$ surface tension $=\left(4 \pi r^{2} n-4 \pi \mathrm{R}^{2}\right) \mathrm{T}$
<br/> <br/>$\left\{4 \pi\left(\frac{\mathrm{R}^{2}}{(8)^{2 / 3}}\right) \times 8-4 \pi \mathrm{R}^{2}\right\} \mathrm{T}=4 \pi \mathrm{R}^{2} \mathrm{~T}$</div>


tags: ['Physics', 'Electrostatics', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-spherically-symmetric-gravitational-system-of-particles-has-a-mass-density-0-0-for-r-r-for-r-r-where-r-0-is-a-constant-a-test-mass-can-undergo-circu-1

slug:
a-spherically-symmetric-gravitational-system-of-particles-has-a-mass-density-0-0-for-r-r-for-r-r-where-r-0-is-a-constant-a-test-mass-can-undergo-circu-1-68232

question:
<div class="question">A spherically symmetric gravitational system of particles has a mass density $\rho=\left\{\begin{array}{ll}\rho_{0} &amp; \text { for } \mathrm{r} \leq \mathrm{R} \\ 0 &amp; \text { for } \mathrm{r}&gt;\mathrm{R}\end{array}\right.$
<br/> <br/>where $\mathrm{r}_{0}$ is a constant. A test mass can undergo
<br/> <br/>circular motion under the influence of the gravitational field of particles. Its speed $\mathrm{V}$ as a function of distance $r(0<r by="" centre="" from="" is="" of="" represented="" system="" the=""> <div class="tags" data-pagefind-ignore=""> <span class="tag">Physics</span><span class="tag">Electrostatics</span><span class="tag">JEE Main</span> </div> <span class="label">Options:</span>
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/4_mVNkW8izvk4YpMzNKAmHSO0bin9AXBKI6-p65iKKc.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/uw-k7WgNaR6afUBiWAXADXwmrGzFD8-hxY-4VHV4RlQ.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ky9ucJjGYT4arSBcXVk96lSrhwvZIwCeMwskrxZAQLY.original.fullsize.png"/></span> </li> </ul> <!-- /Question --> <!-- Solution --> <section class="solution-container" id="accepted-answer"> <div class="solution-header"> <span class="label">Solution:</span> <div class="badges-container" data-pagefind-ignore=""> <span class="badge upvotes-count"> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.625 5.0075C14.4842 4.84795 14.3111 4.72019 14.1171 4.63269C13.9231 4.54519 13.7128 4.49996 13.5 4.5H10V3.5C10 2.83696 9.73661 2.20107 9.26777 1.73223C8.79893 1.26339 8.16304 1 7.5 1C7.40711 0.999934 7.31604 1.02574 7.237 1.07454C7.15795 1.12333 7.09407 1.19318 7.0525 1.27625L4.69125 6H2C1.73478 6 1.48043 6.10536 1.29289 6.29289C1.10536 6.48043 1 6.73478 1 7V12.5C1 12.7652 1.10536 13.0196 1.29289 13.2071C1.48043 13.3946 1.73478 13.5 2 13.5H12.75C13.1154 13.5001 13.4684 13.3668 13.7425 13.1252C14.0166 12.8835 14.1931 12.5501 14.2388 12.1875L14.9888 6.1875C15.0153 5.97626 14.9966 5.76179 14.9339 5.55833C14.8712 5.35488 14.7659 5.16711 14.625 5.0075ZM2 7H4.5V12.5H2V7Z" fill="#014A82"></path> </svg> 1262 Upvotes
</span> <span class="badge expert-verification-badge" data-pagefind-ignore=""> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.1163 6.42625C13.8806 6.18 13.6369 5.92625 13.545 5.70312C13.46 5.49875 13.455 5.16 13.45 4.83187C13.4406 4.22187 13.4306 3.53062 12.95 3.05C12.4694 2.56937 11.7781 2.55937 11.1681 2.55C10.84 2.545 10.5012 2.54 10.2969 2.455C10.0744 2.36312 9.82 2.11937 9.57375 1.88375C9.1425 1.46937 8.6525 1 8 1C7.3475 1 6.85812 1.46937 6.42625 1.88375C6.18 2.11937 5.92625 2.36312 5.70312 2.455C5.5 2.54 5.16 2.545 4.83187 2.55C4.22187 2.55937 3.53062 2.56937 3.05 3.05C2.56937 3.53062 2.5625 4.22187 2.55 4.83187C2.545 5.16 2.54 5.49875 2.455 5.70312C2.36312 5.92562 2.11937 6.18 1.88375 6.42625C1.46937 6.8575 1 7.3475 1 8C1 8.6525 1.46937 9.14187 1.88375 9.57375C2.11937 9.82 2.36312 10.0738 2.455 10.2969C2.54 10.5012 2.545 10.84 2.55 11.1681C2.55937 11.7781 2.56937 12.4694 3.05 12.95C3.53062 13.4306 4.22187 13.4406 4.83187 13.45C5.16 13.455 5.49875 13.46 5.70312 13.545C5.92562 13.6369 6.18 13.8806 6.42625 14.1163C6.8575 14.5306 7.3475 15 8 15C8.6525 15 9.14187 14.5306 9.57375 14.1163C9.82 13.8806 10.0738 13.6369 10.2969 13.545C10.5012 13.46 10.84 13.455 11.1681 13.45C11.7781 13.4406 12.4694 13.4306 12.95 12.95C13.4306 12.4694 13.4406 11.7781 13.45 11.1681C13.455 10.84 13.46 10.5012 13.545 10.2969C13.6369 10.0744 13.8806 9.82 14.1163 9.57375C14.5306 9.1425 15 8.6525 15 8C15 7.3475 14.5306 6.85812 14.1163 6.42625ZM10.8538 6.85375L7.35375 10.3538C7.30731 10.4002 7.25217 10.4371 7.19147 10.4623C7.13077 10.4874 7.06571 10.5004 7 10.5004C6.93429 10.5004 6.86923 10.4874 6.80853 10.4623C6.74783 10.4371 6.69269 10.4002 6.64625 10.3538L5.14625 8.85375C5.05243 8.75993 4.99972 8.63268 4.99972 8.5C4.99972 8.36732 5.05243 8.24007 5.14625 8.14625C5.24007 8.05243 5.36732 7.99972 5.5 7.99972C5.63268 7.99972 5.75993 8.05243 5.85375 8.14625L7 9.29313L10.1462 6.14625C10.1927 6.09979 10.2479 6.06294 10.3086 6.0378C10.3692 6.01266 10.4343 5.99972 10.5 5.99972C10.5657 5.99972 10.6308 6.01266 10.6914 6.0378C10.7521 6.06294 10.8073 6.09979 10.8538 6.14625C10.9002 6.1927 10.9371 6.24786 10.9622 6.30855C10.9873 6.36925 11.0003 6.4343 11.0003 6.5C11.0003 6.5657 10.9873 6.63075 10.9622 6.69145C10.9371 6.75214 10.9002 6.8073 10.8538 6.85375Z" fill="#00672E"></path> </svg>
Verified Answer
</span> </div> </div> <div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> </div> <div class="solution">Given mass density \(P=\left\{P_{0}\right.\) for \(r \leq R O\) for \(\left.r&gt;R\right\}\)
<br/> <br/>Since gravitational depends on mass, which is zero for \(r&gt;R\), hence it only acts for \(r \leq R\).
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\overrightarrow{\mathrm{F}}_{\text {graw }}=\overrightarrow{\mathrm{F}}_{\text {circular }} \\
<br/> <br/>&amp;\Rightarrow \frac{\mathrm{GM}_{\text {sphere }} \times \mathrm{M}}{\mathrm{R}^{2}}=\frac{\mathrm{M}(\mathrm{e})^{2}}{\mathrm{R}} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{M}_{\text {sphere }}}{\mathrm{R}} \text { since } \mathrm{r} \leq \mathrm{R} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{R}^{3}}{\mathrm{R}} \alpha \mathrm{R}^{2} \Rightarrow \mathrm{V} \alpha \mathrm{R} \quad \text { straight line }+\mathrm{ve} \text { slope }
<br/> <br/>\end{aligned}\)
<br/> <br/>Beyond \(\mathrm{r}=\mathrm{R}\), mass density \(=0 \Rightarrow\) acceleration due to gravity \(=0 \Rightarrow\) velocity stays constant.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/gsdhgwt463hsjhwu.png"/></div> </section> <!-- /Solution --> <!-- Call to Action --> <section class="call-to-action" data-pagefind-ignore=""> <div class="call-to-action-header"> <svg fill="none" height="24" viewbox="0 0 25 24" width="25" xmlns="http://www.w3.org/2000/svg"> <path d="M19.9999 13.5C20.0018 13.8058 19.9089 14.1047 19.734 14.3555C19.559 14.6063 19.3107 14.7968 19.023 14.9006L14.1874 16.6875L12.4062 21.5269C12.3007 21.8134 12.1098 22.0608 11.8594 22.2355C11.6089 22.4101 11.3109 22.5038 11.0055 22.5038C10.7002 22.5038 10.4022 22.4101 10.1517 22.2355C9.90124 22.0608 9.71039 21.8134 9.60491 21.5269L7.8124 16.6875L2.97303 14.9062C2.68646 14.8008 2.43915 14.6099 2.26445 14.3595C2.08976 14.109 1.99609 13.811 1.99609 13.5056C1.99609 13.2003 2.08976 12.9022 2.26445 12.6518C2.43915 12.4013 2.68646 12.2105 2.97303 12.105L7.8124 10.3125L9.59366 5.47313C9.69914 5.18656 9.88998 4.93924 10.1404 4.76455C10.3909 4.58985 10.6889 4.49619 10.9943 4.49619C11.2996 4.49619 11.5977 4.58985 11.8481 4.76455C12.0986 4.93924 12.2894 5.18656 12.3949 5.47313L14.1874 10.3125L19.0268 12.0938C19.3146 12.1986 19.5628 12.3901 19.7371 12.642C19.9114 12.8939 20.0033 13.1937 19.9999 13.5ZM14.7499 4.5H16.2499V6C16.2499 6.19891 16.3289 6.38968 16.4696 6.53033C16.6102 6.67098 16.801 6.75 16.9999 6.75C17.1988 6.75 17.3896 6.67098 17.5302 6.53033C17.6709 6.38968 17.7499 6.19891 17.7499 6V4.5H19.2499C19.4488 4.5 19.6396 4.42098 19.7802 4.28033C19.9209 4.13968 19.9999 3.94891 19.9999 3.75C19.9999 3.55109 19.9209 3.36032 19.7802 3.21967C19.6396 3.07902 19.4488 3 19.2499 3H17.7499V1.5C17.7499 1.30109 17.6709 1.11032 17.5302 0.96967C17.3896 0.829018 17.1988 0.75 16.9999 0.75C16.801 0.75 16.6102 0.829018 16.4696 0.96967C16.3289 1.11032 16.2499 1.30109 16.2499 1.5V3H14.7499C14.551 3 14.3602 3.07902 14.2196 3.21967C14.0789 3.36032 13.9999 3.55109 13.9999 3.75C13.9999 3.94891 14.0789 4.13968 14.2196 4.28033C14.3602 4.42098 14.551 4.5 14.7499 4.5ZM22.9999 7.5H22.2499V6.75C22.2499 6.55109 22.1709 6.36032 22.0302 6.21967C21.8896 6.07902 21.6988 6 21.4999 6C21.301 6 21.1102 6.07902 20.9696 6.21967C20.8289 6.36032 20.7499 6.55109 20.7499 6.75V7.5H19.9999C19.801 7.5 19.6102 7.57902 19.4696 7.71967C19.3289 7.86032 19.2499 8.05109 19.2499 8.25C19.2499 8.44891 19.3289 8.63968 19.4696 8.78033C19.6102 8.92098 19.801 9 19.9999 9H20.7499V9.75C20.7499 9.94891 20.8289 10.1397 20.9696 10.2803C21.1102 10.421 21.301 10.5 21.4999 10.5C21.6988 10.5 21.8896 10.421 22.0302 10.2803C22.1709 10.1397 22.2499 9.94891 22.2499 9.75V9H22.9999C23.1988 9 23.3896 8.92098 23.5302 8.78033C23.6709 8.63968 23.7499 8.44891 23.7499 8.25C23.7499 8.05109 23.6709 7.86032 23.5302 7.71967C23.3896 7.57902 23.1988 7.5 22.9999 7.5Z" fill="#1589EE"></path> </svg> <h3>Looking for more such questions to practice?</h3> </div> <p>
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options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/4_mVNkW8izvk4YpMzNKAmHSO0bin9AXBKI6-p65iKKc.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/uw-k7WgNaR6afUBiWAXADXwmrGzFD8-hxY-4VHV4RlQ.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ky9ucJjGYT4arSBcXVk96lSrhwvZIwCeMwskrxZAQLY.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Given mass density \(P=\left\{P_{0}\right.\) for \(r \leq R O\) for \(\left.r&gt;R\right\}\)
<br/> <br/>Since gravitational depends on mass, which is zero for \(r&gt;R\), hence it only acts for \(r \leq R\).
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\overrightarrow{\mathrm{F}}_{\text {graw }}=\overrightarrow{\mathrm{F}}_{\text {circular }} \\
<br/> <br/>&amp;\Rightarrow \frac{\mathrm{GM}_{\text {sphere }} \times \mathrm{M}}{\mathrm{R}^{2}}=\frac{\mathrm{M}(\mathrm{e})^{2}}{\mathrm{R}} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{M}_{\text {sphere }}}{\mathrm{R}} \text { since } \mathrm{r} \leq \mathrm{R} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{R}^{3}}{\mathrm{R}} \alpha \mathrm{R}^{2} \Rightarrow \mathrm{V} \alpha \mathrm{R} \quad \text { straight line }+\mathrm{ve} \text { slope }
<br/> <br/>\end{aligned}\)
<br/> <br/>Beyond \(\mathrm{r}=\mathrm{R}\), mass density \(=0 \Rightarrow\) acceleration due to gravity \(=0 \Rightarrow\) velocity stays constant.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/gsdhgwt463hsjhwu.png"/></div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch1_P1.db
title:
a-spherically-symmetric-gravitational-system-of-particles-has-a-mass-density-0-0-for-r-r-for-r-r-where-r-0-is-a-constant-a-test-mass-can-undergo-circu

slug:
a-spherically-symmetric-gravitational-system-of-particles-has-a-mass-density-0-0-for-r-r-for-r-r-where-r-0-is-a-constant-a-test-mass-can-undergo-circu-74549

question:
<div class="question">A spherically symmetric gravitational system of particles has a mass density $\rho=\left\{\begin{array}{ll}\rho_{0} &amp; \text { for } \mathrm{r} \leq \mathrm{R} \\ 0 &amp; \text { for } \mathrm{r}&gt;\mathrm{R}\end{array}\right.$
<br/> <br/>where $\mathrm{r}_{0}$ is a constant. A test mass can undergo
<br/> <br/>circular motion under the influence of the gravitational field of particles. Its speed $\mathrm{V}$ as a function of distance $r(0<r by="" centre="" from="" is="" of="" represented="" system="" the=""> <div class="tags" data-pagefind-ignore=""> <span class="tag">Physics</span><span class="tag">Electrostatics</span><span class="tag">BITSAT</span><span class="tag">BITSAT 2014</span> </div> <span class="label">Options:</span>
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/4_mVNkW8izvk4YpMzNKAmHSO0bin9AXBKI6-p65iKKc.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/uw-k7WgNaR6afUBiWAXADXwmrGzFD8-hxY-4VHV4RlQ.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ky9ucJjGYT4arSBcXVk96lSrhwvZIwCeMwskrxZAQLY.original.fullsize.png"/></span> </li> </ul> <!-- /Question --> <!-- Solution --> <section class="solution-container" id="accepted-answer"> <div class="solution-header"> <span class="label">Solution:</span> <div class="badges-container" data-pagefind-ignore=""> <span class="badge upvotes-count"> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.625 5.0075C14.4842 4.84795 14.3111 4.72019 14.1171 4.63269C13.9231 4.54519 13.7128 4.49996 13.5 4.5H10V3.5C10 2.83696 9.73661 2.20107 9.26777 1.73223C8.79893 1.26339 8.16304 1 7.5 1C7.40711 0.999934 7.31604 1.02574 7.237 1.07454C7.15795 1.12333 7.09407 1.19318 7.0525 1.27625L4.69125 6H2C1.73478 6 1.48043 6.10536 1.29289 6.29289C1.10536 6.48043 1 6.73478 1 7V12.5C1 12.7652 1.10536 13.0196 1.29289 13.2071C1.48043 13.3946 1.73478 13.5 2 13.5H12.75C13.1154 13.5001 13.4684 13.3668 13.7425 13.1252C14.0166 12.8835 14.1931 12.5501 14.2388 12.1875L14.9888 6.1875C15.0153 5.97626 14.9966 5.76179 14.9339 5.55833C14.8712 5.35488 14.7659 5.16711 14.625 5.0075ZM2 7H4.5V12.5H2V7Z" fill="#014A82"></path> </svg> 1141 Upvotes
</span> <span class="badge expert-verification-badge" data-pagefind-ignore=""> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.1163 6.42625C13.8806 6.18 13.6369 5.92625 13.545 5.70312C13.46 5.49875 13.455 5.16 13.45 4.83187C13.4406 4.22187 13.4306 3.53062 12.95 3.05C12.4694 2.56937 11.7781 2.55937 11.1681 2.55C10.84 2.545 10.5012 2.54 10.2969 2.455C10.0744 2.36312 9.82 2.11937 9.57375 1.88375C9.1425 1.46937 8.6525 1 8 1C7.3475 1 6.85812 1.46937 6.42625 1.88375C6.18 2.11937 5.92625 2.36312 5.70312 2.455C5.5 2.54 5.16 2.545 4.83187 2.55C4.22187 2.55937 3.53062 2.56937 3.05 3.05C2.56937 3.53062 2.5625 4.22187 2.55 4.83187C2.545 5.16 2.54 5.49875 2.455 5.70312C2.36312 5.92562 2.11937 6.18 1.88375 6.42625C1.46937 6.8575 1 7.3475 1 8C1 8.6525 1.46937 9.14187 1.88375 9.57375C2.11937 9.82 2.36312 10.0738 2.455 10.2969C2.54 10.5012 2.545 10.84 2.55 11.1681C2.55937 11.7781 2.56937 12.4694 3.05 12.95C3.53062 13.4306 4.22187 13.4406 4.83187 13.45C5.16 13.455 5.49875 13.46 5.70312 13.545C5.92562 13.6369 6.18 13.8806 6.42625 14.1163C6.8575 14.5306 7.3475 15 8 15C8.6525 15 9.14187 14.5306 9.57375 14.1163C9.82 13.8806 10.0738 13.6369 10.2969 13.545C10.5012 13.46 10.84 13.455 11.1681 13.45C11.7781 13.4406 12.4694 13.4306 12.95 12.95C13.4306 12.4694 13.4406 11.7781 13.45 11.1681C13.455 10.84 13.46 10.5012 13.545 10.2969C13.6369 10.0744 13.8806 9.82 14.1163 9.57375C14.5306 9.1425 15 8.6525 15 8C15 7.3475 14.5306 6.85812 14.1163 6.42625ZM10.8538 6.85375L7.35375 10.3538C7.30731 10.4002 7.25217 10.4371 7.19147 10.4623C7.13077 10.4874 7.06571 10.5004 7 10.5004C6.93429 10.5004 6.86923 10.4874 6.80853 10.4623C6.74783 10.4371 6.69269 10.4002 6.64625 10.3538L5.14625 8.85375C5.05243 8.75993 4.99972 8.63268 4.99972 8.5C4.99972 8.36732 5.05243 8.24007 5.14625 8.14625C5.24007 8.05243 5.36732 7.99972 5.5 7.99972C5.63268 7.99972 5.75993 8.05243 5.85375 8.14625L7 9.29313L10.1462 6.14625C10.1927 6.09979 10.2479 6.06294 10.3086 6.0378C10.3692 6.01266 10.4343 5.99972 10.5 5.99972C10.5657 5.99972 10.6308 6.01266 10.6914 6.0378C10.7521 6.06294 10.8073 6.09979 10.8538 6.14625C10.9002 6.1927 10.9371 6.24786 10.9622 6.30855C10.9873 6.36925 11.0003 6.4343 11.0003 6.5C11.0003 6.5657 10.9873 6.63075 10.9622 6.69145C10.9371 6.75214 10.9002 6.8073 10.8538 6.85375Z" fill="#00672E"></path> </svg>
Verified Answer
</span> </div> </div> <div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> </div> <div class="solution">Given mass density \(P=\left\{P_{0}\right.\) for \(r \leq R O\) for \(\left.r&gt;R\right\}\)
<br/> <br/>Since gravitational depends on mass, which is zero for \(r&gt;R\), hence it only acts for \(r \leq R\).
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\overrightarrow{\mathrm{F}}_{\text {graw }}=\overrightarrow{\mathrm{F}}_{\text {circular }} \\
<br/> <br/>&amp;\Rightarrow \frac{\mathrm{GM}_{\text {sphere }} \times \mathrm{M}}{\mathrm{R}^{2}}=\frac{\mathrm{M}(\mathrm{e})^{2}}{\mathrm{R}} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{M}_{\text {sphere }}}{\mathrm{R}} \text { since } \mathrm{r} \leq \mathrm{R} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{R}^{3}}{\mathrm{R}} \alpha \mathrm{R}^{2} \Rightarrow \mathrm{V} \alpha \mathrm{R} \quad \text { straight line }+\mathrm{ve} \text { slope }
<br/> <br/>\end{aligned}\)
<br/> <br/>Beyond \(\mathrm{r}=\mathrm{R}\), mass density \(=0 \Rightarrow\) acceleration due to gravity \(=0 \Rightarrow\) velocity stays constant.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/gsdhgwt463hsjhwu.png"/></div> </section> <!-- /Solution --> <!-- Call to Action --> <section class="call-to-action" data-pagefind-ignore=""> <div class="call-to-action-header"> <svg fill="none" height="24" viewbox="0 0 25 24" width="25" xmlns="http://www.w3.org/2000/svg"> <path d="M19.9999 13.5C20.0018 13.8058 19.9089 14.1047 19.734 14.3555C19.559 14.6063 19.3107 14.7968 19.023 14.9006L14.1874 16.6875L12.4062 21.5269C12.3007 21.8134 12.1098 22.0608 11.8594 22.2355C11.6089 22.4101 11.3109 22.5038 11.0055 22.5038C10.7002 22.5038 10.4022 22.4101 10.1517 22.2355C9.90124 22.0608 9.71039 21.8134 9.60491 21.5269L7.8124 16.6875L2.97303 14.9062C2.68646 14.8008 2.43915 14.6099 2.26445 14.3595C2.08976 14.109 1.99609 13.811 1.99609 13.5056C1.99609 13.2003 2.08976 12.9022 2.26445 12.6518C2.43915 12.4013 2.68646 12.2105 2.97303 12.105L7.8124 10.3125L9.59366 5.47313C9.69914 5.18656 9.88998 4.93924 10.1404 4.76455C10.3909 4.58985 10.6889 4.49619 10.9943 4.49619C11.2996 4.49619 11.5977 4.58985 11.8481 4.76455C12.0986 4.93924 12.2894 5.18656 12.3949 5.47313L14.1874 10.3125L19.0268 12.0938C19.3146 12.1986 19.5628 12.3901 19.7371 12.642C19.9114 12.8939 20.0033 13.1937 19.9999 13.5ZM14.7499 4.5H16.2499V6C16.2499 6.19891 16.3289 6.38968 16.4696 6.53033C16.6102 6.67098 16.801 6.75 16.9999 6.75C17.1988 6.75 17.3896 6.67098 17.5302 6.53033C17.6709 6.38968 17.7499 6.19891 17.7499 6V4.5H19.2499C19.4488 4.5 19.6396 4.42098 19.7802 4.28033C19.9209 4.13968 19.9999 3.94891 19.9999 3.75C19.9999 3.55109 19.9209 3.36032 19.7802 3.21967C19.6396 3.07902 19.4488 3 19.2499 3H17.7499V1.5C17.7499 1.30109 17.6709 1.11032 17.5302 0.96967C17.3896 0.829018 17.1988 0.75 16.9999 0.75C16.801 0.75 16.6102 0.829018 16.4696 0.96967C16.3289 1.11032 16.2499 1.30109 16.2499 1.5V3H14.7499C14.551 3 14.3602 3.07902 14.2196 3.21967C14.0789 3.36032 13.9999 3.55109 13.9999 3.75C13.9999 3.94891 14.0789 4.13968 14.2196 4.28033C14.3602 4.42098 14.551 4.5 14.7499 4.5ZM22.9999 7.5H22.2499V6.75C22.2499 6.55109 22.1709 6.36032 22.0302 6.21967C21.8896 6.07902 21.6988 6 21.4999 6C21.301 6 21.1102 6.07902 20.9696 6.21967C20.8289 6.36032 20.7499 6.55109 20.7499 6.75V7.5H19.9999C19.801 7.5 19.6102 7.57902 19.4696 7.71967C19.3289 7.86032 19.2499 8.05109 19.2499 8.25C19.2499 8.44891 19.3289 8.63968 19.4696 8.78033C19.6102 8.92098 19.801 9 19.9999 9H20.7499V9.75C20.7499 9.94891 20.8289 10.1397 20.9696 10.2803C21.1102 10.421 21.301 10.5 21.4999 10.5C21.6988 10.5 21.8896 10.421 22.0302 10.2803C22.1709 10.1397 22.2499 9.94891 22.2499 9.75V9H22.9999C23.1988 9 23.3896 8.92098 23.5302 8.78033C23.6709 8.63968 23.7499 8.44891 23.7499 8.25C23.7499 8.05109 23.6709 7.86032 23.5302 7.71967C23.3896 7.57902 23.1988 7.5 22.9999 7.5Z" fill="#1589EE"></path> </svg> <h3>Looking for more such questions to practice?</h3> </div> <p>
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options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/4_mVNkW8izvk4YpMzNKAmHSO0bin9AXBKI6-p65iKKc.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/uw-k7WgNaR6afUBiWAXADXwmrGzFD8-hxY-4VHV4RlQ.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ky9ucJjGYT4arSBcXVk96lSrhwvZIwCeMwskrxZAQLY.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zlmjtZrHwk9bZI1kJjha9uzSrXfRvbMccZpIFYZiThM.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Given mass density \(P=\left\{P_{0}\right.\) for \(r \leq R O\) for \(\left.r&gt;R\right\}\)
<br/> <br/>Since gravitational depends on mass, which is zero for \(r&gt;R\), hence it only acts for \(r \leq R\).
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\overrightarrow{\mathrm{F}}_{\text {graw }}=\overrightarrow{\mathrm{F}}_{\text {circular }} \\
<br/> <br/>&amp;\Rightarrow \frac{\mathrm{GM}_{\text {sphere }} \times \mathrm{M}}{\mathrm{R}^{2}}=\frac{\mathrm{M}(\mathrm{e})^{2}}{\mathrm{R}} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{M}_{\text {sphere }}}{\mathrm{R}} \text { since } \mathrm{r} \leq \mathrm{R} \\
<br/> <br/>&amp;\Rightarrow \mathrm{V}^{2} \alpha \frac{\mathrm{R}^{3}}{\mathrm{R}} \alpha \mathrm{R}^{2} \Rightarrow \mathrm{V} \alpha \mathrm{R} \quad \text { straight line }+\mathrm{ve} \text { slope }
<br/> <br/>\end{aligned}\)
<br/> <br/>Beyond \(\mathrm{r}=\mathrm{R}\), mass density \(=0 \Rightarrow\) acceleration due to gravity \(=0 \Rightarrow\) velocity stays constant.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/gsdhgwt463hsjhwu.png"/></div>


tags: ['Physics', 'Gravitation', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-spherically-symmetric-gravitational-systom-of-particles-has-mass-density-0-for-r-r-0-for-r-r-where-0-is-a-constant-a-test-mass-can-undergo-circular--1

slug:
a-spherically-symmetric-gravitational-systom-of-particles-has-mass-density-0-for-r-r-0-for-r-r-where-0-is-a-constant-a-test-mass-can-undergo-circular-1-27400

question:
<div class="question"><p>A spherically symmetric gravitational systom of particles has mass density <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi><mo>=</mo><mfenced close="" open="{"><mtable columnalign="left"><mtr><mtd><msub><mi>ρ</mi><mn>0</mn></msub><mtext> for </mtext><mi>r</mi><mo>≤</mo><mi>R</mi></mtd></mtr><mtr><mtd><mn>0</mn><mtext> for </mtext><mi>r</mi><mo>&gt;</mo><mi>R</mi></mtd></mtr></mtable></mfenced></math> where, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mn>0</mn></msub></math> is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> as a function of distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>(</mo><mn>0</mn><mo>&lt;</mo><mi>r</mi><mo>&lt;</mo><mo>∞</mo><mo>)</mo></math> from the centre of the system is represented by</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ee9959d4-c3de-4b7e-a650-b7177c2b466e-image.png" style="width: 200px; height: 199px;"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5bdfa4e5-1fa4-48aa-9367-d42bb5fc7a9e-image.png" style="width: 200px; height: 186px;"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f3d51331-af17-47e0-8711-e46893c7b554-image.png" style="width: 200px; height: 192px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2fce4ae9-45e4-4e84-bc1c-3ece4cf1970b-image.png" style="width: 200px; height: 196px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f3d51331-af17-47e0-8711-e46893c7b554-image.png" style="width: 200px; height: 192px;"/></span> </div>

answer:
<div class="solution"><p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≤</mo><mi>R</mi></math>, then force on the test mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> at the surface of the sphere <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>m</mi><mi>g</mi></math></p><p>Force on the test mass at distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> from the centre of sphere is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/eceb191e-a1ed-4170-868c-67d4c4662a07-image.png" style="width: 200px; height: 226px;"/></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>&lt;</mo><mi>R</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>R</mi><mn>2</mn></msup></mfrac><mo>·</mo><mfrac><mi>r</mi><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>R</mi><mn>3</mn></msup></mfrac><mo>·</mo><mi>r</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><mrow><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>R</mi><mn>3</mn></msup></mfrac><mo>·</mo><mi>r</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi>v</mi><mo> </mo><mi>α</mi><mo> </mo><mi>r</mi></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>&gt;</mo><mi>R</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><mrow><mi>M</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>v</mi><mo> </mo><mi>α</mi><mo> </mo><mfrac><mn>1</mn><msqrt><mi>r</mi></msqrt></mfrac></math></p></div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-spherically-symmetric-gravitational-systom-of-particles-has-mass-density-0-for-r-r-0-for-r-r-where-0-is-a-constant-a-test-mass-can-undergo-circular-

slug:
a-spherically-symmetric-gravitational-systom-of-particles-has-mass-density-0-for-r-r-0-for-r-r-where-0-is-a-constant-a-test-mass-can-undergo-circular-12325

question:
<div class="question"><p>A spherically symmetric gravitational systom of particles has mass density <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi><mo>=</mo><mfenced close="" open="{"><mtable columnalign="left"><mtr><mtd><msub><mi>ρ</mi><mn>0</mn></msub><mtext> for </mtext><mi>r</mi><mo>≤</mo><mi>R</mi></mtd></mtr><mtr><mtd><mn>0</mn><mtext> for </mtext><mi>r</mi><mo>&gt;</mo><mi>R</mi></mtd></mtr></mtable></mfenced></math> where, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mn>0</mn></msub></math> is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> as a function of distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>(</mo><mn>0</mn><mo>&lt;</mo><mi>r</mi><mo>&lt;</mo><mo>∞</mo><mo>)</mo></math> from the centre of the system is represented by</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ee9959d4-c3de-4b7e-a650-b7177c2b466e-image.png" style="width: 200px; height: 199px;"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5bdfa4e5-1fa4-48aa-9367-d42bb5fc7a9e-image.png" style="width: 200px; height: 186px;"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f3d51331-af17-47e0-8711-e46893c7b554-image.png" style="width: 200px; height: 192px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2fce4ae9-45e4-4e84-bc1c-3ece4cf1970b-image.png" style="width: 200px; height: 196px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f3d51331-af17-47e0-8711-e46893c7b554-image.png" style="width: 200px; height: 192px;"/></span> </div>

answer:
<div class="solution"><p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≤</mo><mi>R</mi></math>, then force on the test mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> at the surface of the sphere <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>m</mi><mi>g</mi></math></p><p>Force on the test mass at distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> from the centre of sphere is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/eceb191e-a1ed-4170-868c-67d4c4662a07-image.png" style="width: 200px; height: 226px;"/></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>&lt;</mo><mi>R</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>R</mi><mn>2</mn></msup></mfrac><mo>·</mo><mfrac><mi>r</mi><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>R</mi><mn>3</mn></msup></mfrac><mo>·</mo><mi>r</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><mrow><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>R</mi><mn>3</mn></msup></mfrac><mo>·</mo><mi>r</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi>v</mi><mo> </mo><mi>α</mi><mo> </mo><mi>r</mi></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>&gt;</mo><mi>R</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><mrow><mi>M</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac><mo>=</mo><mfrac><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>v</mi><mo> </mo><mi>α</mi><mo> </mo><mfrac><mn>1</mn><msqrt><mi>r</mi></msqrt></mfrac></math></p></div>


title: a-spontaneous-reaction-is-impossible-if
slug: a-spontaneous-reaction-is-impossible-if-99208
question: <div class="question">A spontaneous reaction is impossible if</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch1_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">both $\Delta \mathrm{H}$ and $\Delta \mathrm{S}$ are negative</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">both $\Delta \mathrm{H}$ and $\Delta \mathrm{S}$ are positive</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\Delta \mathrm{H}$ is negative and $\Delta \mathrm{S}$ is positive</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\Delta \mathrm{H}$ is positive and $\Delta \mathrm{S}$ is negative</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\Delta \mathrm{H}$ is positive and $\Delta \mathrm{S}$ is negative</span> </div>

answer:
<div class="solution">$\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \Delta \mathrm{S} ; \Delta \mathrm{G}$ is positive for a reaction
<br/> <br/>to be non-spontaneous when $\Delta \mathrm{H}$ is positive and $\Delta \mathrm{S}$ is negative.</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch1_P1.db
title:
a-spring-is-compressed-between-two-toy-carts-of-mass-m-1-and-m-2-when-the-toy-carts-are-released-the-springs-exert-equal-and-opposite-average-forces-f

slug:
a-spring-is-compressed-between-two-toy-carts-of-mass-m-1-and-m-2-when-the-toy-carts-are-released-the-springs-exert-equal-and-opposite-average-forces-f-70871

question:
<div class="question">A spring is compressed between two toy carts of mass $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$. When the toy carts are released, the springs exert equal and opposite average forces for the same time on each toy cart. If $v_{1}$ and $\mathrm{v}_{2}$ are the velocities of the toy carts and there is no friction between the toy carts and the ground, then :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{v}_{1} / \mathrm{v}_{2}=\mathrm{m}_{1} / \mathrm{m}_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{v}_{1} / \mathrm{v}_{2}=\mathrm{m}_{2} / \mathrm{m}_{1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{v}_{1} / \mathrm{v}_{2}=-\mathrm{m}_{2} / \mathrm{m}_{1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{v}_{1} / \mathrm{v}_{2}=-\mathrm{m}_{1} / \mathrm{m}_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{v}_{1} / \mathrm{v}_{2}=-\mathrm{m}_{2} / \mathrm{m}_{1}$</span> </div>

answer:
<div class="solution">Applying law of conservation of linear momentum<br/>$\mathrm{m}_{1} \mathrm{v}_{1}+\mathrm{m}_{2} \mathrm{v}_{2}=0$<br/>$\frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}=-\frac{\mathrm{v}_{2}}{\mathrm{v}_{1}}$ or $\frac{\mathrm{v}_{1}}{\mathrm{v}_{2}}=-\frac{\mathrm{m}_{2}}{\mathrm{~m}_{1}}$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch1_P1.db
title:
a-square-gate-of-size-1-m-1-m-is-hinged-at-its-midpoint-a-fluid-of-density-fills-the-space-to-the-left-of-the-gate-the-force-f-required-to-hold-the-ga

slug:
a-square-gate-of-size-1-m-1-m-is-hinged-at-its-midpoint-a-fluid-of-density-fills-the-space-to-the-left-of-the-gate-the-force-f-required-to-hold-the-ga-83564

question:
<div class="question">A square gate of size $1 \mathrm{~m} \times 1 \mathrm{~m}$ is hinged at its mid-point. A fluid of density $\rho$ fills the space to the left of the gate. The force F required to hold the gate stationary is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/XVQ6n8l4hIPD-N4kpGu5JUES9xX1RTV_u98VibumF5s.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\rho g}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\rho g}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\rho g}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\rho g}{8}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\rho g}{6}$</span> </div>

answer:
<div class="solution">The net force acting on the gate element of width dy at a depth y from the surface of the fluid, is
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\mathrm{dy} &amp;=\left(\mathrm{p}_{0}+\rho_{\mathrm{g}} \mathrm{y}-\mathrm{p}_{0}\right) \times 1 \mathrm{dy} \\
<br/> <br/>&amp;=\rho \mathrm{gydy}
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>Torque about the hinge is
<br/> <br/>$$
<br/> <br/>\mathrm{d} \tau=\operatorname{pgydy} \times\left(\frac{1}{2}-\mathrm{y}\right)
<br/> <br/>$$
<br/> <br/>Net torque experience. by the gate is
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\tau_{\text {net }} &amp;=\int \mathrm{d} \tau+\mathrm{F} \times \frac{1}{2} \\
<br/> <br/>&amp;=\int_{0}^{1} \rho g \mathrm{ydy}\left(\frac{1}{2}-\mathrm{y}\right)+\mathrm{F} \times \frac{1}{2}=0 \\
<br/> <br/>\Rightarrow &amp; \mathrm{F}=\frac{\rho \mathrm{g}}{6}
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>i.e., The force $F$ required to hold the gate
<br/> <br/>stationary is $\frac{p g}{a}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e9c12zlBUuQI3hCrHSf5mHlJePI2pdgm1s0nk4-zrfE.original.fullsize.png"/></div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
answer: <div class="solution">Bysymmetry, the magnetic field at the centre Pis zero.</div>
source_db: MarksBatch1_P1.db
title:
a-steady-current-is-set-up-in-a-cubic-network-composed-of-wires-of-equal-resistance-and-length-d-as-shown-in-figure-what-is-the-magnetic-field-at-the-

slug:
a-steady-current-is-set-up-in-a-cubic-network-composed-of-wires-of-equal-resistance-and-length-d-as-shown-in-figure-what-is-the-magnetic-field-at-the-84166

question:
<div class="question">A steady current is set up in a cubic network composed of wires of equal resistance and length d as shown in figure. What is the magnetic field at the centre $\mathrm{P}$ due to the cubic network<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/JlynlWJ2-d8PqEIJ4Pe8qvzv8rTuZ8GfBJgTxi5Pmi0.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mu_{0}}{4 \pi} \frac{2 I}{d}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mu_{0}}{4 \pi} \frac{2 \mathrm{I}}{\sqrt{2} \mathrm{~d}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mu_{0}}{4 \pi} \frac{\theta \pi \mathrm{I}}{\mathrm{d}}$</span> </li> </ul>


tags: ['Physics', 'Mechanical Properties of Solids', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch1_P1.db
title:
a-steel-wire-of-crosssectional-area-4-cm-2-has-elastic-limit-of-2-2-10-8-n-m-2-the-maximum-upward-acceleration-that-can-be-given-to-a-1000-kg-elevator

slug:
a-steel-wire-of-crosssectional-area-4-cm-2-has-elastic-limit-of-2-2-10-8-n-m-2-the-maximum-upward-acceleration-that-can-be-given-to-a-1000-kg-elevator-96889

question:
<div class="question">A steel wire of cross-sectional area <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><msup><mi>cm</mi><mn>2</mn></msup></math> has elastic limit of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo> </mo><mi mathvariant="normal">N</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>. The maximum upward acceleration that can be given to a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn><mo> </mo><mi>kg</mi></math> elevator supported by this steel wire if the stress is to exceed one-fourth of the elastic limit is [Take, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>]</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p><p>Area of cross-section of steel wire,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>4</mn><mo> </mo><msup><mi>cm</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup></math></p><p>Mass of elevator, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>1000</mn><mo> </mo><mi>kg</mi></math></p><p>The maximum tension in the steel wire that can support elevator is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>×</mo><mtext> stress </mtext><mo>×</mo><mtext> area of cross-section </mtext></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>×</mo><mn>2</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo>×</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo> </mo><mi mathvariant="normal">N</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> be the maximum upward acceleration of the elevator, then</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mi>m</mi><mo> </mo><mfenced><mrow><mi>g</mi><mo>+</mo><mi>f</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo>=</mo><mn>1000</mn><mo> </mo><mfenced><mrow><mn>10</mn><mo>+</mo><mi>f</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo> </mo><mi>f</mi><mo>=</mo><mn>12</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p></div>


tags: ['Physics', 'Motion In One Dimension', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-stone-is-projected-with-velocity-2-g-h-so-that-it-just-clears-two-walls-of-equal-height-h-at-distance-of-2-h-from-each-other-the-time-interval-of-pa

slug:
a-stone-is-projected-with-velocity-2-g-h-so-that-it-just-clears-two-walls-of-equal-height-h-at-distance-of-2-h-from-each-other-the-time-interval-of-pa-14632

question:
<div class="question">A stone is projected with velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mi>g</mi><mi>h</mi></msqrt></math>, so that it just clears two walls of equal height <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, at distance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>h</mi></math> from each other. The time interval of passing between the two walls is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mi>h</mi><mi>g</mi></mfrac></msqrt></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mn>2</mn><mi>h</mi></mrow><mi>g</mi></mfrac></msqrt></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mfrac><mi>h</mi><mi>g</mi></mfrac></msqrt></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>h</mi></mrow><mi>g</mi></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mfrac><mi>h</mi><mi>g</mi></mfrac></msqrt></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>h</mi><mo>=</mo><mi>u</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mi>h</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo> </mo><mfenced close="]" open="["><mrow><mo>∵</mo><mi>u</mi><mo>=</mo><msub><mi>u</mi><mi>x</mi></msub><mo>,</mo><mi>t</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msub><mi>u</mi><mi>y</mi></msub><mi>t</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>g</mi><msup><mi>t</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msub><mi>u</mi><mi>y</mi></msub><mi>t</mi><mo>+</mo><mn>2</mn><mi>h</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>2</mn><msub><mi>u</mi><mi>y</mi></msub><mo>+</mo><msqrt><mn>4</mn><msubsup><mi>u</mi><mi>y</mi><mn>2</mn></msubsup><mo>-</mo><mn>8</mn><mi>g</mi><mi>h</mi></msqrt></mrow><mrow><mn>2</mn><mi>g</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><mn>2</mn><msub><mi>u</mi><mi>y</mi></msub><mo>-</mo><msqrt><mn>4</mn><msubsup><mi>u</mi><mi>y</mi><mn>2</mn></msubsup><mo>-</mo><mn>8</mn><mi>g</mi><mi>h</mi></msqrt></mrow><mrow><mn>2</mn><mi>g</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>=</mo><msub><mi>t</mi><mn>1</mn></msub><mo>-</mo><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><msqrt><mn>4</mn><msubsup><mi>u</mi><mi>y</mi><mn>2</mn></msubsup><mo>-</mo><mn>8</mn><mi>g</mi><mi>h</mi></msqrt><mi>g</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>u</mi><mi>y</mi><mn>2</mn></msubsup><mo>=</mo><mfrac><mrow><msup><mi>g</mi><mn>2</mn></msup><mo>(</mo><mi>Δ</mi><mi>t</mi><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mn>2</mn><mi>g</mi><mi>h</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>u</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>u</mi><mi>y</mi><mn>2</mn></msubsup><mo>=</mo><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><mo>(</mo><mn>2</mn><msqrt><mi>g</mi><mi>h</mi></msqrt><msup><mo>)</mo><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup></mrow><mrow><mo>(</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>g</mi><mn>2</mn></msup><mo>(</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>+</mo><mn>2</mn><mi>g</mi><mi>h</mi><mo>=</mo><mn>4</mn><mi>g</mi><mi>h</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>g</mi><mn>2</mn></msup><mn>4</mn></mfrac><msup><mfenced><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced><mn>4</mn></msup><mo>-</mo><mn>2</mn><mi>g</mi><mi>h</mi><msup><mfenced><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>g</mi><mi>h</mi><mo>±</mo><msqrt><mn>4</mn><msup><mi>g</mi><mn>2</mn></msup><msup><mi>h</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>g</mi><mn>2</mn></msup><msup><mi>h</mi><mn>2</mn></msup></msqrt></mrow><mrow><msup><mi>g</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mi>h</mi></mrow><mi>g</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>=</mo><mn>2</mn><msqrt><mfrac><mi>h</mi><mi>g</mi></mfrac></msqrt></math></p></div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch1_P1.db
title:
a-stone-projected-with-a-velocity-a-at-an-angle-with-the-horizontal-reaches-maximum-height-h-1-when-it-is-projected-with-velocity-u-at-an-angle-2-with

slug:
a-stone-projected-with-a-velocity-a-at-an-angle-with-the-horizontal-reaches-maximum-height-h-1-when-it-is-projected-with-velocity-u-at-an-angle-2-with-87675

question:
<div class="question">A stone projected with a velocity a at an angle $\theta$ with the horizontal reaches maximum height $\mathrm{H}_{1}$. When it is projected with velocity u at an angle $\left(\frac{\pi}{2}-\theta\right)$ with the horizontal, it reaches maximum height $\mathrm{H}_{2}$. The relation between the horizontal range $\mathrm{R}$ of the projectile, heights $\mathrm{H}_{1}$ and $\mathrm{H}_{2}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{R}=4 \sqrt{\mathrm{H}_{1} \mathrm{H}_{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{R}=4\left(\mathrm{H}_{1}-\mathrm{H}_{2}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{R}=4\left(\mathrm{H}_{1}+\mathrm{H}_{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{R}=\frac{\mathrm{H}_{1}^{2}}{\mathrm{H}_{2}^{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{R}=4 \sqrt{\mathrm{H}_{1} \mathrm{H}_{2}}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{H}_{1}=\frac{\mathrm{u}^{2} \sin ^{2} \theta}{2 \mathrm{~g}} \\ \text { and } \mathrm{H}_{2}=\frac{\mathrm{u}^{2} \sin ^{2}\left(90^{\circ}-\theta\right)}{2 \mathrm{~g}}=\frac{\mathrm{u}^{2} \cos ^{2} \theta}{2 \mathrm{~g}} \\ \mathrm{H}_{1} \mathrm{H}_{2}=&amp; \frac{u^{2} \sin ^{2} \theta}{2 \mathrm{~g}} \times \frac{u^{2} \cos ^{2} \theta}{2 \mathrm{~g}}=\frac{\left(u^{2} \sin 2 \theta\right)^{2}}{16 g^{2}}=\frac{R^{2}}{16} \\ \therefore R=4 \sqrt{H_{1} H_{2}} \end{aligned}$</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch1_P1.db
title:
a-straight-rod-of-length-9-units-slides-with-its-endsa-b-always-on-the-x-and-yaxis-respectively-then-the-locus-of-the-centroid-of-oab-is

slug:
a-straight-rod-of-length-9-units-slides-with-its-endsa-b-always-on-the-x-and-yaxis-respectively-then-the-locus-of-the-centroid-of-oab-is-27071

question:
<div class="question">A straight rod of length 9 units slides with its endsA, $B$ always on the $X$ and Y-axis respectively. Then the locus of the centroid of $\Delta \mathrm{OAB}$ is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}=3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}=9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}=81$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}=9$</span> </div>

answer:
<div class="solution">Let end $A$ is $(a, 0)$ and end $\mathrm{B}$ is $(0,$ b) then
<br/> <br/>$\mathrm{a}^{2}+\mathrm{b}^{2}=81$
<br/> <br/>If centroid is $(x, y)$ then
<br/> <br/>$\mathrm{x}=\frac{\mathrm{a}}{3}, \mathrm{y}=\frac{\mathrm{b}}{3},$ putting in (1) we get the locus as
<br/> <br/>$x^{2}+y^{2}=9$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Qqa9x5fnBLF9s_aag1z5_Q-eLcCfjsV-5sBp3_CTl2g.original.fullsize.png"/></div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch1_P1.db
title:
a-string-of-length-is-fixed-at-both-ends-it-is-vibrating-in-its-3-rd-overtone-with-maximum-amplitudea-the-amplitude-at-a-distance-3-from-one-end-is

slug:
a-string-of-length-is-fixed-at-both-ends-it-is-vibrating-in-its-3-rd-overtone-with-maximum-amplitudea-the-amplitude-at-a-distance-3-from-one-end-is-51774

question:
<div class="question">A string of length $\ell$ is fixed at both ends. It is vibrating in its $3^{\text {rd }}$ overtone with maximum amplitude'a'. The amplitude at a distance $\ell / 3$ from one end is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3} \mathrm{a}}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{a}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3} \mathrm{a}}{2}$</span> </div>

answer:
<div class="solution">For a string vibrating in its $\mathrm{n}^{\text {th }}$ overtone $(\mathrm{n}+1)^{\text {th }}$ harmonic) $\mathrm{y}=2 \mathrm{~A} \sin \left(\frac{(\mathrm{n}+1) \pi \mathrm{x}}{\mathrm{L}}\right) \cos \omega \mathrm{t}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/q2ItQTJMVJyy48Raj7wmHRjcncF8oiXrcCY1kNpGo3E.original.fullsize.png"/><br/> <br/> <br/>$$
<br/> <br/>\begin{array}{l} 
<br/> <br/>\text { For } \left.x=\frac{\ell}{3}\right] 2 A=a \&amp; n=3 \\
<br/> <br/>y=\left[a \sin \left(\frac{4 \pi}{\ell} \cdot \frac{\ell}{3}\right)\right] \cos \omega t \\
<br/> <br/>=a \sin \frac{4 \pi}{3} \cos \omega t=-a\left(\frac{\sqrt{3}}{2}\right) \cos \omega t
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>i.e. at $x=\frac{\ell}{3},$ the amplitude is $\frac{\sqrt{3} a}{2}$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">35</span> </div>
source_db: MarksBatch1_P1.db
title:
a-student-read-common-difference-of-an-a-p-as-2-instead-of-2-and-got-the-sum-of-first-5-terms-as-5-actual-sum-of-first-five-terms-is

slug:
a-student-read-common-difference-of-an-a-p-as-2-instead-of-2-and-got-the-sum-of-first-5-terms-as-5-actual-sum-of-first-five-terms-is-60682

question:
<div class="question">A student read common difference of an A. P. as
<br/> <br/>-2 instead of 2 and got the sum of first 5 terms as -5 . Actual sum of first five terms is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">25</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-25</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-35</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">35</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\frac{5}{2}\{2 \mathrm{a}+4(-2)\}=-5 \Rightarrow 2 \mathrm{a}=6$
<br/> <br/>$\therefore$ Actual sum $=\frac{5}{2}\{6+4(2)\}=35$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P1.db
title:
a-submarine-a-travelling-at-17-m-s-is-being-chased-along-the-line-of-its-velocity-by-another-submarine-b-travelling-at-34-m-s-b-sends-a-sonar-signal-o

slug:
a-submarine-a-travelling-at-17-m-s-is-being-chased-along-the-line-of-its-velocity-by-another-submarine-b-travelling-at-34-m-s-b-sends-a-sonar-signal-o-59695

question:
<div class="question">A submarine $A$ travelling at $17 \mathrm{~m} / \mathrm{s}$ is being chased along the line of its velocity by another submarine $B$ travelling at $34 \mathrm{~m} / \mathrm{s}$. B sends a sonar signal of $600 \mathrm{~Hz}$ to detect $A$ and receives a reflected sound of frequency $v$. The of $v$ is
<br/>[Speed of sound in water $=1500 \mathrm{~ms}^{-1}$ ]</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$613.7 \mathrm{~Hz}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$6137 \mathrm{~Hz}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$62 \mathrm{~Hz}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$539 \mathrm{~Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$613.7 \mathrm{~Hz}$
<br/></span> </div>

answer:
<div class="solution">$\therefore$ Given, velocity of submarine $A, v_A=17 \mathrm{~m} / \mathrm{s}$
<br/>Velocity of submarine $B, v_B=34 \mathrm{~m} / \mathrm{s}$
<br/>Signal sent by submarine $B$ is detected by submarine $A$ can be shown as
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Jq4hW_ynZTPjtqs3fQMJASQbe7HfYMizbGu9xW3UZps.original.fullsize.png"/><br/> <br/>Freqency of the signal, $f_0=600 \mathrm{~Hz}$
<br/>Speed of sound in water $v_s=1500 \mathrm{~ms}^{-1}$
<br/>Frequency received by submarine $A$ is
<br/>$$
<br/>f_1=\left(\frac{v_{\mathrm{S}}-v_A}{v_{\mathrm{S}}-v_B}\right) f_0=\left[\frac{1500-17}{1500-34}\right] \times 600 \mathrm{~Hz}
<br/>$$
<br/>$$
<br/>f_1=\frac{1483}{1466} \times 600
<br/>$$
<br/>Frequency received by submarine $B$ is
<br/>$$
<br/>f_2=\left(\frac{v_{\mathrm{S}}+v_B}{v_s+v_A}\right) f_1
<br/>$$
<br/>Substituting given values and $f_1$ value from Eq. (i), we get
<br/>$$
<br/>\begin{aligned}
<br/>f_2 &amp; =\left(\frac{1500+34}{1500+17}\right) \times\left[\frac{1483}{1466} \times 600\right] \\
<br/>f_2 &amp; =1.0112 \times 1.0115 \times 600 \\
<br/>f_2 &amp; =613.7 \mathrm{~Hz}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(2,2)</span> </div>
source_db: MarksBatch1_P1.db
title:
a-system-consists-of-three-particles-each-of-mass-m-and-located-at-1122-and-33-the-coordinates-of-the-centre-of-mass-are

slug:
a-system-consists-of-three-particles-each-of-mass-m-and-located-at-1122-and-33-the-coordinates-of-the-centre-of-mass-are-46407

question:
<div class="question">A system consists of three particles, each of mass $\mathrm{m}$ and located at (1,1),(2,2) and (3,3) . The co-ordinates of the centre of mass are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(1,1)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">(2,2)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(3,3)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(6,6)</span> </li> </ul>

answer:
<div class="solution">The coordinates of C.M of three particle are<br/>\begin{array}{c}<br/>\mathrm{x}=\frac{\mathrm{m}_{1} \mathrm{x}_{1}+\mathrm{m}_{2} \mathrm{x}_{2}+\mathrm{m}_{3} \mathrm{x}_{3}}{\mathrm{~m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}} \&amp; \mathrm{y}=\frac{\mathrm{m}_{1} \mathrm{y}_{1}+\mathrm{m}_{2} \mathrm{y}_{2}+\mathrm{m}_{3} \mathrm{y}_{3}}{\mathrm{~m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}} \\<br/>\text { here } \mathrm{m}_{1}=\mathrm{m}_{2}=\mathrm{m}_{3}=\mathrm{m} \\<br/>\text { so } \mathrm{x}=\frac{\left(\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}\right) \mathrm{m}}{\mathrm{m}+\mathrm{m}+\mathrm{m}}=2<br/>\end{array}<br/>$$<br/>y=\frac{\left(y_{1}+y_{2}+y_{3}\right) m}{m+m+m}=2<br/>$$<br/>so coordinates of $\mathrm{C} . \mathrm{M}$. of three particle are (2,2)<br/>$\mathrm{V}=300 \mathrm{~m} / \mathrm{s}$</div>


tags: ['Mathematics', 'Hyperbola', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-tangent-drawn-to-hyperbola-x-2-a-2-y-2-b-2-1-at-p-6-forms-a-triangle-of-area-3-a-2-square-units-with-coordinate-axes-if-the-eccentricity-of-hyperbol

slug:
a-tangent-drawn-to-hyperbola-x-2-a-2-y-2-b-2-1-at-p-6-forms-a-triangle-of-area-3-a-2-square-units-with-coordinate-axes-if-the-eccentricity-of-hyperbol-36450

question:
<div class="question">A tangent drawn to hyperbola <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mfrac><mi>π</mi><mn>6</mn></mfrac></mfenced></math> forms a triangle of area <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup></math> square units, with coordinate axes. If the eccentricity of hyperbola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math>, then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> </div>

answer:
<div class="solution"><p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mfrac><mi>π</mi><mn>6</mn></mfrac></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mi>sec</mi><mfrac><mi>π</mi><mn>6</mn></mfrac><mo>,</mo><mi>b</mi><mi>tan</mi><mfrac><mi>π</mi><mn>6</mn></mfrac></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><msqrt><mn>3</mn></msqrt></mfrac><mo>,</mo><mfrac><mi>b</mi><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Equation of tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mfrac><mrow><msqrt><mn>3</mn></msqrt><mi>a</mi></mrow><mn>2</mn></mfrac></mfrac><mo>-</mo><mfrac><mi>y</mi><mrow><msqrt><mn>3</mn></msqrt><mi>b</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math>1$</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Area of the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mi>a</mi></mrow><mn>2</mn></mfrac><mo>×</mo><msqrt><mn>3</mn></msqrt><mi>b</mi><mo>=</mo><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mi>b</mi><mi>a</mi></mfrac><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>17</mn></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mo>=</mo><mn>17</mn><mo>-</mo><mn>9</mn><mo>=</mo><mn>8</mn></math></p><p>Hence, option (d) is correct.</p></div>


title: a-thermodynamic-system-is-taken-through-cyclic-process-the-total-work-done-in-the-process-is-1
tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch1_P1.db
slug:
a-thermodynamic-system-is-taken-through-cyclic-process-the-total-work-done-in-the-process-is-1-57540

question:
<div class="question">A thermodynamic system is taken through cyclic process. The total work done in the process is :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/r_k-pTakueAeath94qEiCmm7MqVo46295nTfgzwVCcw.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \mathrm{~J}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$300 \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Zero</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$200 \mathrm{~J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$300 \mathrm{~J}$</span> </div>

answer:
<div class="solution">Work done $=$ Area under the curve<br/>$\Rightarrow \mathrm{W}=\frac{1}{2} \times(4-2) \times(400-100)=\frac{1}{2}(2) \times 300$<br/>$\mathrm{W}=300 \mathrm{~J}$</div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch1_P1.db
title:
a-thermodynamical-system-is-changed-from-state-p-1-v-1-to-p-2-v-2-by-two-different-process-the-quantity-which-will-remain-same-will-be

slug:
a-thermodynamical-system-is-changed-from-state-p-1-v-1-to-p-2-v-2-by-two-different-process-the-quantity-which-will-remain-same-will-be-20014

question:
<div class="question">A thermodynamical system is changed from state $\left(\mathrm{P}_{1}, \mathrm{~V}_{1}\right)$ to $\left(\mathrm{P}_{2}, \mathrm{~V}_{2}\right)$ by two different process, the quantity which will remain same will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\Delta \mathrm{Q}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\Delta \mathrm{W}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\Delta \mathrm{Q}+\Delta \mathrm{W}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\Delta \mathrm{Q}-\Delta \mathrm{W}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\Delta \mathrm{Q}-\Delta \mathrm{W}$</span> </div>

answer:
<div class="solution">For all process $\Delta \mathrm{U}=\Delta \mathrm{Q}-\Delta \mathrm{W}$
<br/> <br/>does not change as it depends on initial final states.</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch1_P1.db
title:
a-thin-convergent-glass-lens-g-15-has-a-power-of-50-d-when-this-lens-is-immersed-in-a-liquid-of-refractive-index-it-acts-as-a-divergent-lens-of-focal-

slug:
a-thin-convergent-glass-lens-g-15-has-a-power-of-50-d-when-this-lens-is-immersed-in-a-liquid-of-refractive-index-it-acts-as-a-divergent-lens-of-focal-35117

question:
<div class="question">A thin convergent glass lens $\left(\mu_{g}=1.5\right)$ has a power of $+5.0 \mathrm{D}$. When this lens is immersed in a liquid of refractive index $\mu,$ it acts as a divergent lens of focal length $100 \mathrm{~cm}$. The value of $\mu$ must be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 / 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5 / 3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$6 / 5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 / 3$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{P}_{\mathrm{a}}}{\mathrm{P}_{1}}=\frac{\left(\frac{\mu_{\mathrm{g}}}{\mu_{\mathrm{a}}}-1\right)}{\left(\frac{\mu_{\mathrm{g}}}{\mu_{1}}-1\right)}=\frac{+5}{-100 / 100}=-5$<br/>$-5\left(\frac{\mu_{\mathrm{g}}}{\mu_{1}}-1\right)=\frac{\mu_{\mathrm{g}}}{\mu_{\mathrm{a}}}-1$<br/>$\frac{1.5}{\mu_{1}}-1=\frac{-1}{5}(1.5-1)=-0.1$<br/>$$<br/>\begin{array}{ll}\mu_{1}=\frac{1.5}{0.9}=\frac{5}{3}\end{array}<br/>$$</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch1_P1.db
title:
a-thin-prism-p-1-with-an-angle-6-and-made-of-glass-of-refractive-index-154-is-combined-with-another-prism-p-2-made-from-glass-of-refractive-index-172-

slug:
a-thin-prism-p-1-with-an-angle-6-and-made-of-glass-of-refractive-index-154-is-combined-with-another-prism-p-2-made-from-glass-of-refractive-index-172-49299

question:
<div class="question">A thin prism $P_1$ with an angle $6^{\circ}$ and made of glass of refractive index 1.54 is combined with another prism $\mathrm{P}_2$ made from glass of refractive index 1.72 to produce dispersion without average deviation. The angle of prism $\mathrm{P}_2$ is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.3^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$7.8^{\circ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4.5^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4.5^{\circ}$</span> </div>

answer:
<div class="solution">Angle of deviation for first prism<br/>$\delta_1=A_1\left(\mu_1-1\right)=6(1.54-1)$<br/>Angle of deviation for second prism<br/>$\begin{aligned}<br/>&amp; \delta_2=A_2\left(\mu_2-1\right) \\<br/>&amp; =A_2(1.72-1)<br/>\end{aligned}$<br/>For dispersion without deviation<br/>$\begin{aligned}<br/>&amp; \delta_1=\delta_2 \\<br/>&amp; \Rightarrow 6^{\circ}(1.54-1)=\mathrm{A}_2(1.72-1) \\<br/>&amp; \Rightarrow \mathrm{A}_2=\frac{6^{\circ} \times 0.54}{0.72}=\frac{18^{\circ}}{4}=4.5^{\circ}<br/>\end{aligned}$</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-thin-rod-of-length-41-and-mass-4-m-is-bent-at-the-points-as-shown-in-figure-what-is-the-moment-of-inertia-of-the-rod-about-the-axis-passes-through-p-1

slug:
a-thin-rod-of-length-41-and-mass-4-m-is-bent-at-the-points-as-shown-in-figure-what-is-the-moment-of-inertia-of-the-rod-about-the-axis-passes-through-p-1-82436

question:
<div class="question">A thin rod of length 41 and mass $4 \mathrm{~m}$ is bent at the points as shown in figure. What is the moment of inertia of the rod about the axis passes through point $\mathrm{O}$ and perpendicular to the plane of paper?
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Le_TQucLG1G48_m7oD_lKGohuAK-fnoJ4w4-Wc0mfo4.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{M} l^{2}}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{10 \mathrm{M} l^{2}}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{M} l^{2}}{12}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{M} l^{2}}{24}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{10 \mathrm{M} l^{2}}{3}$</span> </div>

answer:
<div class="solution">Total moment of inertia
<br/> <br/>$$
<br/> <br/>=\mathrm{I}_{1}+\mathrm{I}_{2}+\mathrm{I}_{3}+\mathrm{I}_{4}=2 \mathrm{I}_{1}+2 \mathrm{I}_{2}
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=2\left(l_{1}+l_{2}\right)\left\lceil I_{3}=I_{1}, I_{1}=I_{4}\right] \\
<br/> <br/>\text { Now, } I_{2}=I_{3}=\frac{M I^{2}}{3}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Using parallel axes theorem, we have
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{I}=\mathrm{I}_{\mathrm{CM}}+\mathrm{Mx}^{2} \text { and } \mathrm{x}=\sqrt{l^{2}+\frac{l^{2}}{4}} \\
<br/> <br/>\mathrm{I}_{1}=\mathrm{I}_{4}=\frac{\mathrm{M} l^{2}}{12}+\mathrm{M}\left[\sqrt{l^{2}+\left(\frac{l}{2}\right)^{2}}\right]^{2}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Putting all values we get Moment of inertia, $\mathrm{I}=10\left(\frac{\mathrm{Ml}^{2}}{3}\right)$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch1_P1.db
title:
a-thin-rod-of-length-41-and-mass-4-m-is-bent-at-the-points-as-shown-in-figure-what-is-the-moment-of-inertia-of-the-rod-about-the-axis-passes-through-p

slug:
a-thin-rod-of-length-41-and-mass-4-m-is-bent-at-the-points-as-shown-in-figure-what-is-the-moment-of-inertia-of-the-rod-about-the-axis-passes-through-p-39492

question:
<div class="question">A thin rod of length 41 and mass $4 \mathrm{~m}$ is bent at the points as shown in figure. What is the moment of inertia of the rod about the axis passes through point $\mathrm{O}$ and perpendicular to the plane of paper?
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Le_TQucLG1G48_m7oD_lKGohuAK-fnoJ4w4-Wc0mfo4.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{M} l^{2}}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{10 \mathrm{M} l^{2}}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{M} l^{2}}{12}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{M} l^{2}}{24}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{10 \mathrm{M} l^{2}}{3}$</span> </div>

answer:
<div class="solution">Total moment of inertia
<br/> <br/>$$
<br/> <br/>=\mathrm{I}_{1}+\mathrm{I}_{2}+\mathrm{I}_{3}+\mathrm{I}_{4}=2 \mathrm{I}_{1}+2 \mathrm{I}_{2}
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=2\left(l_{1}+l_{2}\right)\left\lceil I_{3}=I_{1}, I_{1}=I_{4}\right] \\
<br/> <br/>\text { Now, } I_{2}=I_{3}=\frac{M I^{2}}{3}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Using parallel axes theorem, we have
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{I}=\mathrm{I}_{\mathrm{CM}}+\mathrm{Mx}^{2} \text { and } \mathrm{x}=\sqrt{l^{2}+\frac{l^{2}}{4}} \\
<br/> <br/>\mathrm{I}_{1}=\mathrm{I}_{4}=\frac{\mathrm{M} l^{2}}{12}+\mathrm{M}\left[\sqrt{l^{2}+\left(\frac{l}{2}\right)^{2}}\right]^{2}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Putting all values we get Moment of inertia, $\mathrm{I}=10\left(\frac{\mathrm{Ml}^{2}}{3}\right)$</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch1_P1.db
title:
a-thin-sheet-of-glass-15-of-thickness-6-micron-introduced-in-the-path-of-one-of-the-interfering-beams-in-a-double-slit-experiment-shifts-the-central-f

slug:
a-thin-sheet-of-glass-15-of-thickness-6-micron-introduced-in-the-path-of-one-of-the-interfering-beams-in-a-double-slit-experiment-shifts-the-central-f-89995

question:
<div class="question">A thin sheet of glass $(\mu=1.5)$ of thickness 6 micron introduced in the path of one of the interfering beams in a double slit experiment shifts the central fringe to a position previously occupied by fifth bright fringe. Then the wavelength of light used is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$6000 Ã$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3000 Ã$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4500 Ã$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7500 Ã$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6000 Ã$</span> </div>

answer:
<div class="solution">$\mathrm{n} \lambda=(\mu-1) \mathrm{t} ;$
<br/> <br/>$\therefore \lambda=\frac{(\mu-1) \mathrm{t}}{\mathrm{n}}=\frac{(1.5-1) \times 6 \times 10^{-6}}{5}=6000 \mathrm{~A}$</div>


tags: ['Physics', 'Motion In One Dimension', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch1_P1.db
title:
a-train-accelerating-uniformly-from-rest-attains-a-maximum-speed-of-40-ms-1-in-20-s-it-travels-at-the-speed-for-20-s-and-is-brought-to-rest-with-unifo

slug:
a-train-accelerating-uniformly-from-rest-attains-a-maximum-speed-of-40-ms-1-in-20-s-it-travels-at-the-speed-for-20-s-and-is-brought-to-rest-with-unifo-44197

question:
<div class="question">A train accelerating uniformly from rest attains a maximum speed of $40 \mathrm{~ms}^{-1}$ in $20 \mathrm{~s}$. It travels at the speed for $20 \mathrm{~s}$ and is brought to rest with uniform retardation in further $40 \mathrm{~s}$. What is the average velocity during the period ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$80 \mathrm{~m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$25 \mathrm{~m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$40 \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$30 \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$25 \mathrm{~m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">(i) $\mathrm{v}=\mathrm{u}+$ at $_{1}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>40=0+a \times 20 \\
<br/> <br/>a=2 m / s^{2} \\
<br/> <br/>v^{2}-u^{2}=2 a s \\
<br/> <br/>40^{2}-0=2 \times 2 s_{1} \\
<br/> <br/>\text { (ii) } s_{1}=400 \mathrm{~m} \\
<br/> <br/>\text { (iii) } v=u+a t \\
<br/> <br/>\quad 0=40+a \times 40 \\
<br/> <br/>a=-1 \mathrm{~m} / \mathrm{s}^{2} \\
<br/> <br/>0^{2}-40^{2}=2(-1) \mathrm{s}_{3} \\
<br/> <br/>\mathrm{~s}_{3}=800 \mathrm{~m} \\
<br/> <br/>\text { Total distance travelled }=\mathrm{s}_{1}+\mathrm{s}_{2}+\mathrm{s}_{3} \\
<br/> <br/>=400+800+800=2000 \mathrm{~m} \\
<br/> <br/>\text { Total time taken }=20+20+40=80 \mathrm{~s} \\
<br/> <br/>\text { Average velocity }=\frac{2000}{80}=25 \mathrm{~m} / \mathrm{s}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-train-moves-towards-a-stationary-observer-with-speed-34-m-s-1-the-train-blows-whistle-and-its-frequency-is-registered-by-the-observer-as-1-if-the-tr

slug:
a-train-moves-towards-a-stationary-observer-with-speed-34-m-s-1-the-train-blows-whistle-and-its-frequency-is-registered-by-the-observer-as-1-if-the-tr-62094

question:
<div class="question">A train moves towards a stationary observer with speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>34</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. The train blows whistle and its frequency is registered by the observer as <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ν</mi><mn>1</mn></msub></math>. If the train's speed is reduced to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo></math> then the frequency registered is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ν</mi><mn>2</mn></msub></math>. If the speed of sound is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>340</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, then the ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>ν</mi><mn>1</mn></msub><msub><mi>ν</mi><mn>2</mn></msub></mfrac></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>19</mn><mn>18</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>18</mn><mn>19</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>/</mo><mn>2</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>19</mn><mn>18</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>According to Doppler's effect, the approximate frequency heard by the stationary observer,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ν</mi><mo>=</mo><mfrac><mi>C</mi><mrow><mi>C</mi><mo>-</mo><msub><mi>v</mi><mi>s</mi></msub></mrow></mfrac><msub><mi>ν</mi><mn>0</mn></msub></math></p><p>Case (I) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>s</mi></msub><mo>=</mo><mn>34</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo></math>Speed of sound in air,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>s</mi></msub><mo>=</mo></math>Speed of source, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ν</mi><mn>0</mn></msub><mo>=</mo></math> frequency of the source.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo>  </mo><msub><mi>ν</mi><mn>1</mn></msub><mo>=</mo><mfenced><mfrac><mn>340</mn><mrow><mn>340</mn><mo>-</mo><mn>34</mn></mrow></mfrac></mfenced><msub><mi>ν</mi><mn>0</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>340</mn><mn>306</mn></mfrac><msub><mi>ν</mi><mrow><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mrow></msub><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>Case (ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>s</mi></msub><mo>=</mo><mn>17</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo> </mo><mo> </mo><msub><mi>ν</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>340</mn><mrow><mn>340</mn><mo>-</mo><mn>17</mn></mrow></mfrac><msub><mi>ν</mi><mn>0</mn></msub><mo>=</mo><mfrac><mn>340</mn><mn>323</mn></mfrac><msub><mi>ν</mi><mn>0</mn></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>From Eq. (i) and (ii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo> </mo><mo> </mo><mfrac><msub><mi>ν</mi><mn>1</mn></msub><msub><mi>ν</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><mrow><mn>340</mn><mo>/</mo><mn>306</mn></mrow><mrow><mn>340</mn><mo>/</mo><mn>323</mn></mrow></mfrac><mo>=</mo><mfrac><mn>323</mn><mn>306</mn></mfrac><mo>=</mo><mfrac><mn>19</mn><mn>18</mn></mfrac></math></p></div>


title: a-transistor-has-a-base-current-of-1-ma-and-emitter-current-90-ma-the-collector-current-will
slug: a-transistor-has-a-base-current-of-1-ma-and-emitter-current-90-ma-the-collector-current-will-63586
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch1_P1.db
question:
<div class="question">A transistor has a base current of $1 \mathrm{~mA}$ and emitter current $90 \mathrm{~mA}$. The collector current will</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$90 \mathrm{~mA}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 \mathrm{~mA}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$89 \mathrm{~mA}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$91 \mathrm{~mA}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$89 \mathrm{~mA}$</span> </div>

answer:
<div class="solution">$\quad \mathrm{I}_{\mathrm{C}}=\mathrm{I}_{\mathrm{E}}-\mathrm{I}_{\mathrm{B}}=90-1=89 \mathrm{~mA}$</div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P1.db
title:
a-transistor-is-connected-in-commonemitter-ce-configuration-the-collector-supply-is-8-v-and-the-voltage-drop-across-a-resistor-is-500-in-the-collector

slug:
a-transistor-is-connected-in-commonemitter-ce-configuration-the-collector-supply-is-8-v-and-the-voltage-drop-across-a-resistor-is-500-in-the-collector-18497

question:
<div class="question">A transistor is connected in common-emitter (CE) configuration. The collector supply is $8 \mathrm{~V}$ and the voltage drop across a resistor is $500 \Omega$ in the collector circuit is $0.6 \mathrm{~V}$. If the current gain factor $\alpha$ is $0.96$, find the base current</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$25 \mu \mathrm{A}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$50 \mu \mathrm{A}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$20 \mu \mathrm{A}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$35 \mu \mathrm{A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$50 \mu \mathrm{A}$
<br/></span> </div>

answer:
<div class="solution">Given, $V_{C C}=8 \mathrm{~V}$
<br/>Voltage drop, $V_0=\Delta I_C R_L=0.6 \mathrm{~V}$
<br/>Load Resistor, $R_L=500 \Omega$
<br/>Gain factor, $\alpha=0.96$
<br/>$\because \quad \beta=\frac{\alpha}{1-\alpha}=\frac{0.96}{1-0.96}=24$
<br/>Now, $\quad V_0=\Delta I_C R_L$
<br/>$0.6=\Delta I_C \times 500$
<br/>$\Delta I_C=\frac{0.6}{500}=1.2 \times 10^{-3} \mathrm{~A}$
<br/>Now,
<br/>$\beta=\frac{\Delta I_C}{\Delta I_B}$
<br/>$\therefore$ Base current $\Delta I_B=\frac{\Delta I_C}{\beta}=\frac{1.2 \times 10^{-3}}{24}=5 \times 10^{-5}$
<br/>$=50 \times 10^{-6}$
<br/>$\Delta I_B=50 \mu \mathrm{A}$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch1_P1.db
title:
a-transverse-wave-propagating-on-a-stretched-string-of-linear-density-3-10-4-kgm-1-is-represented-by-the-equation-y-0-2-sin-1-5-x-60-t-where-x-in-metr

slug:
a-transverse-wave-propagating-on-a-stretched-string-of-linear-density-3-10-4-kgm-1-is-represented-by-the-equation-y-0-2-sin-1-5-x-60-t-where-x-in-metr-12814

question:
<div class="question"><p>A transverse wave propagating on a stretched string of linear density <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><msup><mi>kgm</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> is represented by the equation</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>sin</mi><mo>(</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>60</mn><mi>t</mi><mo>)</mo></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in metres and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is in seconds. The tension in the string (in Newton) is</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>24</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>48</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>20</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>80</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>48</mn></math></span> </div>

answer:
<div class="solution"><p>Equation of wave, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>sin</mi><mo>[</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>60</mn><mi>t</mi><mo>]</mo></math></p><p>Comparing with standard equation, we get </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo>[</mo><mi>k</mi><mi>x</mi><mo>+</mo><mi>ω</mi><mi>t</mi><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>λ</mi></mfrac></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>60</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>T</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Velocity of wave, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mi>ω</mi><mi>k</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>60</mn><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mn>40</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Velocity of wave in stretched string, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msqrt><mfrac><mi>T</mi><mi>m</mi></mfrac></msqrt></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is the linear density, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is tension in the string.</p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mi>m</mi><mo>=</mo><mo>(</mo><mn>40</mn><msup><mo>)</mo><mn>2</mn></msup><mo>×</mo><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>48</mn><mo> </mo><mi mathvariant="normal">N</mi></math></p></div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch1_P1.db
title:
a-trolley-having-mass-of-200-kg-moves-with-uniform-speed-of-36-kmh-1-on-a-frictionless-track-a-child-of-mass-20-kg-runs-on-the-trolley-from-one-end-to

slug:
a-trolley-having-mass-of-200-kg-moves-with-uniform-speed-of-36-kmh-1-on-a-frictionless-track-a-child-of-mass-20-kg-runs-on-the-trolley-from-one-end-to-93112

question:
<div class="question">A trolley having mass of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn><mo> </mo><mi>kg</mi></math> moves with uniform speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn><mo> </mo><msup><mi>kmh</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> on a frictionless track. A child of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mi>kg</mi></math> runs on the trolley from one end to the other (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi></math> away)  with a speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> relative to the trolley in a direction opposite to its inotion and ultimately jumps out of the trolley. With how much velocity has the trolley moved from the time the child begins to run?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>.</mo><mn>36</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo>.</mo><mn>36</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>.</mo><mn>36</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn><mo>.</mo><mn>40</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>.</mo><mn>36</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><p>since, no external force is acting on the system, we can apply conservation of linear momentum.</p><p>Speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn><mo> </mo><mi>kg</mi></math> trolley <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>36</mn><mo>×</mo><mn>1000</mn></mrow><mrow><mn>60</mn><mo>×</mo><mn>60</mn></mrow></mfrac><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> be the initial velocity of trolley, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>b</mi></msub></math> be the absolute velocity of the boy after the beginning of journey of the boy, their relative velocity is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>v</mi><mo>'</mo></msup><mo>-</mo><msub><mi>v</mi><mi>b</mi></msub><mo>=</mo><mn>4</mn><mo>⇒</mo><msub><mi>v</mi><mi>b</mi></msub><mo>=</mo><mfenced><mrow><msup><mi>v</mi><mo>'</mo></msup><mo>-</mo><mn>4</mn></mrow></mfenced></math></p><p>Now, applying law of conservation of momentum,</p><p>Momentum before the boy begins tu run<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> Momentum after the beginning of boy's running</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mn>220</mn><mo>×</mo><mn>10</mn><mo>=</mo><mn>200</mn><msup><mi>v</mi><mo>'</mo></msup><mo>+</mo><mn>20</mn><mfenced><mrow><msup><mi>v</mi><mo>'</mo></msup><mo>-</mo><mn>4</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2200</mn><mo>=</mo><mn>220</mn><msup><mi>v</mi><mo>'</mo></msup><mo>-</mo><mn>80</mn><mo>⇒</mo><mn>220</mn><msup><mi>v</mi><mo>'</mo></msup><mo>=</mo><mn>2280</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>v</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>2280</mn><mn>220</mn></mfrac><mo>=</mo><mn>10</mn><mo>.</mo><mn>36</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p></div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch1_P1.db
title:
a-tunnel-has-been-dug-through-the-centre-of-the-earth-and-a-ball-is-released-in-it-it-executes-shm-with-time-period

slug:
a-tunnel-has-been-dug-through-the-centre-of-the-earth-and-a-ball-is-released-in-it-it-executes-shm-with-time-period-80516

question:
<div class="question">A tunnel has been dug through the centre of the earth and a ball is released in it. It executes S.H.M. with time period</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">42 minutes</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 day</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 hour</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">84.6 minutes</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">84.6 minutes</span> </div>

answer:
<div class="solution">$\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{R}}{\mathrm{g}}}=2 \pi \sqrt{\frac{64 \times 10^{6}}{9.8}}$
<br/> <br/>$=2 \times \frac{22}{7} \times \frac{8 \times 10^{3}}{7 \times \sqrt{2}}$
<br/> <br/>$=\frac{\sqrt{2} \times 22 \times 8 \times 1000}{49 \times 60} \min =84.6 \mathrm{~min}$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
title:
a-u-shaped-tube-contains-a-liquid-of-density-rho-and-it-is-rotated-about-the-left-dotted-line-as-shown-in-the-figure-find-the-difference-in-the-levels

slug:
a-u-shaped-tube-contains-a-liquid-of-density-rho-and-it-is-rotated-about-the-left-dotted-line-as-shown-in-the-figure-find-the-difference-in-the-levels-61364

question:
<div class="question">A U - shaped tube contains a liquid of density rho and it is rotated about the left dotted line as shown in the figure. Find the difference in the levels of liquid column.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/itbBl74FkucEKGG7QzZvFPY0cLivC7hEVSxWFXGFi0M.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\omega^{2} \mathrm{~L}^{2}}{2 g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\omega^{2} L^{2}}{2 \sqrt{2} g}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2 \omega^{2} \mathrm{~L}^{2}}{g}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2 \sqrt{2} \omega^{2} \mathrm{~L}^{2}}{g}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\omega^{2} \mathrm{~L}^{2}}{2 g}$</span> </div>

answer:
<div class="solution">Weight of liquid of height \(\mathrm{H}=\frac{\pi \mathrm{d}^{2}}{4} \times \mathrm{H} \rho \times \mathrm{g} \rightarrow\) (i)
<br/> <br/>Let us consider a mass dm situated at a distance \(x\) from \(A\) as shown in the figure. The centripetal force required for the mass to rotate \(=(\mathrm{dm}) \omega^{2}\)
<br/> <br/>\(\therefore\) The total centripetal force required for the mass of length \(L\) to rotate \(=\int_{0}^{1}\left(\rho \mathrm{x} \frac{\pi \mathrm{d}^{2}}{4}\right) \mathrm{xdx} \times\left(\mathrm{x} \omega^{2}\right)=\rho \times \frac{\pi \mathrm{d}^{2}}{4} \times \omega^{2} \times \frac{1^{2}}{2} \rightarrow\) (ii)
<br/> <br/>This centripetal force required is provided by the weight of liquid of height 'H' From (i) and (ii)
<br/> <br/>\(\begin{aligned} &amp; \frac{\pi \mathrm{d}^{2}}{4} \times \mathrm{H} \times \rho \times \mathrm{g}=\rho \times \frac{\pi \mathrm{d}^{2}}{4} \times \omega^{2} \times \frac{1^{2}}{2} \\ &amp; \Rightarrow \mathrm{H}=\frac{\omega^{2} \mathrm{l}^{2}}{2 \mathrm{~g}} \end{aligned}\)
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/sdhgeeut3473.png"/></div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-uniform-cube-of-side-a-and-mass-m-rests-on-a-rough-horizontal-table-a-horizontal-force-f-is-applied-normal-to-one-of-the-faces-at-a-point-that-is-di-1

slug:
a-uniform-cube-of-side-a-and-mass-m-rests-on-a-rough-horizontal-table-a-horizontal-force-f-is-applied-normal-to-one-of-the-faces-at-a-point-that-is-di-1-82903

question:
<div class="question">A uniform cube of side a and mass m rests on a rough horizontal table. A horizontal force $\mathrm{F}$ is applied normal to one of the faces at a point that is directly above the centre of the face, at a height $3 \mathrm{a} / 4$ above the base. The minimum value of $\mathrm{F}$ for which the cube begins to topple an edge is (assume that cube does not slide)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/YrfzD_XOL40MULVsPnTe2iSYEtgjwGgq5EaxY862HdA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{mg}}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{mg}}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{2 \mathrm{mg}}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3 \mathrm{mg}}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 \mathrm{mg}}{3}$</span> </div>

answer:
<div class="solution">For toppling about edge $x$ x At the moment of toppling the normal force pass through axis $\mathrm{xx}$ '.<br/>$$<br/>\mathrm{F}_{\min } \frac{3 \mathrm{a}}{4}=\mathrm{mg} \frac{\mathrm{a}}{2} \text { or } \mathrm{F}_{\min }=\frac{2 \mathrm{mg}}{3}<br/>$$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch1_P1.db
title:
a-uniform-cube-of-side-a-and-mass-m-rests-on-a-rough-horizontal-table-a-horizontal-force-f-is-applied-normal-to-one-of-the-faces-at-a-point-that-is-di

slug:
a-uniform-cube-of-side-a-and-mass-m-rests-on-a-rough-horizontal-table-a-horizontal-force-f-is-applied-normal-to-one-of-the-faces-at-a-point-that-is-di-36470

question:
<div class="question">A uniform cube of side a and mass m rests on a rough horizontal table. A horizontal force $\mathrm{F}$ is applied normal to one of the faces at a point that is directly above the centre of the face, at a height $3 \mathrm{a} / 4$ above the base. The minimum value of $\mathrm{F}$ for which the cube begins to topple an edge is (assume that cube does not slide)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/YrfzD_XOL40MULVsPnTe2iSYEtgjwGgq5EaxY862HdA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{mg}}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{mg}}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{2 \mathrm{mg}}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3 \mathrm{mg}}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 \mathrm{mg}}{3}$</span> </div>

answer:
<div class="solution">For toppling about edge $x$ x At the moment of toppling the normal force pass through axis $\mathrm{xx}$ '.<br/>$$<br/>\mathrm{F}_{\min } \frac{3 \mathrm{a}}{4}=\mathrm{mg} \frac{\mathrm{a}}{2} \text { or } \mathrm{F}_{\min }=\frac{2 \mathrm{mg}}{3}<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch1_P1.db
title:
a-uniform-electric-field-and-uniform-magnetic-field-are-acting-along-the-same-direction-in-a-certain-region-if-an-electron-is-projected-in-the-region-

slug:
a-uniform-electric-field-and-uniform-magnetic-field-are-acting-along-the-same-direction-in-a-certain-region-if-an-electron-is-projected-in-the-region-46151

question:
<div class="question">A uniform electric field and uniform magnetic field are acting along the same direction in a certain region. If an electron is projected in the region such that its velocity is pointed along the direction of fields, then the electron</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">will turn towards right of direction of motion</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">speed will decrease</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">speed will increase</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">will turn towards left direction of motion</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">speed will decrease</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathrm{v}}$ and $\overrightarrow{\mathrm{B}}$ are in same direction so that magnetic force on electron becomes zero, only electric force acts. But force on electron due to electric field is opposite to the direction of velocity.</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-uniform-rod-of-length-l-and-mass-m-is-free-to-rotate-in-a-vertical-plane-about-a-the-rod-initially-in-horizontal-position-is-released-the-initial-an-2

slug:
a-uniform-rod-of-length-l-and-mass-m-is-free-to-rotate-in-a-vertical-plane-about-a-the-rod-initially-in-horizontal-position-is-released-the-initial-an-2-88033

question:
<div class="question"><p>A uniform rod of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> and mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is free to rotate in a vertical plane about <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>. The rod initially in horizontal position is released. The initial angular acceleration of the rod is (Moment of inertia of rod about <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>m</mi><msup><mi>l</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac></math>)</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/94ef2144-e0eb-46d5-955b-554beab9130e-image.png" style="width: 320px; height: 97px;"/></p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mn>3</mn><mi>g</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><msup><mi>l</mi><mn>2</mn></msup></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>g</mi><mfrac><mi>l</mi><mn>2</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p>The moment of inertia of the uniform rod about an axis through one end and perpendicular to its length is </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>l</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is the mass and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> its length.</p><p>Torque <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>τ</mi><mo>=</mo><mi>I</mi><mi>α</mi></mrow></mfenced></math> acting on centre of gravity of rod is given by </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo>=</mo><mi>m</mi><mi>g</mi><mfrac><mi>l</mi><mn>2</mn></mfrac></math></p><p>also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>m</mi><msup><mi>l</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac><mo>·</mo><mi>α</mi><mo>=</mo><mi>m</mi><mi>g</mi><mfrac><mi>l</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math></p></div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch1_P1.db
title:
a-uniform-rod-of-length-l-and-mass-m-is-free-to-rotate-in-a-vertical-plane-about-a-the-rod-initially-in-horizontal-position-is-released-the-initial-an

slug:
a-uniform-rod-of-length-l-and-mass-m-is-free-to-rotate-in-a-vertical-plane-about-a-the-rod-initially-in-horizontal-position-is-released-the-initial-an-56347

question:
<div class="question"><p>A uniform rod of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> and mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is free to rotate in a vertical plane about <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>. The rod initially in horizontal position is released. The initial angular acceleration of the rod is (Moment of inertia of rod about <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>m</mi><msup><mi>l</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac></math>)</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/94ef2144-e0eb-46d5-955b-554beab9130e-image.png" style="width: 320px; height: 97px;"/></p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mn>3</mn><mi>g</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><msup><mi>l</mi><mn>2</mn></msup></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>g</mi><mfrac><mi>l</mi><mn>2</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p>The moment of inertia of the uniform rod about an axis through one end and perpendicular to its length is </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>l</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is the mass and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> its length.</p><p>Torque <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>τ</mi><mo>=</mo><mi>I</mi><mi>α</mi></mrow></mfenced></math> acting on centre of gravity of rod is given by </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo>=</mo><mi>m</mi><mi>g</mi><mfrac><mi>l</mi><mn>2</mn></mfrac></math></p><p>also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>m</mi><msup><mi>l</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac><mo>·</mo><mi>α</mi><mo>=</mo><mi>m</mi><mi>g</mi><mfrac><mi>l</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>g</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math></p></div>


